1337xyz1337xyz/leandojo-benchmark4-v10 huggingface dataset:
Theorem: MeasureTheory.AnalyticSet.measurableSet_of_compl
File: Mathlib/MeasureTheory/Constructions/Polish.lean

tactic: rcases hs.measurablySeparable hsc disjoint_compl_right with ⟨u, hsu, hdu, hmu⟩
state_before: α : Type u_1
ι : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
s : Set α
hs : AnalyticSet s
hsc : AnalyticSet sᶜ
⊢ MeasurableSet s
state_after: case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
s : Set α
hs : AnalyticSet s
hsc : AnalyticSet sᶜ
u : Set α
hsu : s ⊆ u
hdu : Disjoint sᶜ u
hmu : MeasurableSet u
⊢ MeasurableSet s

tactic: obtain rfl : s = u := hsu.antisymm (disjoint_compl_left_iff_subset.1 hdu)
state_before: case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
s : Set α
hs : AnalyticSet s
hsc : AnalyticSet sᶜ
u : Set α
hsu : s ⊆ u
hdu : Disjoint sᶜ u
hmu : MeasurableSet u
⊢ MeasurableSet s
state_after: case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
s : Set α
hs : AnalyticSet s
hsc : AnalyticSet sᶜ
hsu : s ⊆ s
hdu : Disjoint sᶜ s
hmu : MeasurableSet s
⊢ MeasurableSet s

tactic: exact hmu
state_before: case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
s : Set α
hs : AnalyticSet s
hsc : AnalyticSet sᶜ
hsu : s ⊆ s
hdu : Disjoint sᶜ s
hmu : MeasurableSet s
⊢ MeasurableSet s
state_after: no goals
Theorem: NumberField.mixedEmbedding.finrank
File: Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean

tactic: classical
rw [finrank_prod, finrank_pi, finrank_pi_fintype, Complex.finrank_real_complex, sum_const,
  card_univ, ← NrRealPlaces, ← NrComplexPlaces, ← card_real_embeddings, Algebra.id.smul_eq_mul,
  mul_comm, ← card_complex_embeddings, ← NumberField.Embeddings.card K ℂ,
  Fintype.card_subtype_compl, Nat.add_sub_of_le (Fintype.card_subtype_le _)]
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ finrank ℝ (({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)) = finrank ℚ K
state_after: no goals

tactic: rw [finrank_prod, finrank_pi, finrank_pi_fintype, Complex.finrank_real_complex, sum_const,
  card_univ, ← NrRealPlaces, ← NrComplexPlaces, ← card_real_embeddings, Algebra.id.smul_eq_mul,
  mul_comm, ← card_complex_embeddings, ← NumberField.Embeddings.card K ℂ,
  Fintype.card_subtype_compl, Nat.add_sub_of_le (Fintype.card_subtype_le _)]
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ finrank ℝ (({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)) = finrank ℚ K
state_after: no goals
Theorem: Asymptotics.isBigOWith_zero'
File: Mathlib/Analysis/Asymptotics/Asymptotics.lean

tactic: simp
state_before: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
E' : Type u_6
F' : Type u_7
G' : Type u_8
E'' : Type u_9
F'' : Type u_10
G'' : Type u_11
E''' : Type u_12
R : Type u_13
R' : Type u_14
𝕜 : Type u_15
𝕜' : Type u_16
inst✝¹³ : Norm E
inst✝¹² : Norm F
inst✝¹¹ : Norm G
inst✝¹⁰ : SeminormedAddCommGroup E'
inst✝⁹ : SeminormedAddCommGroup F'
inst✝⁸ : SeminormedAddCommGroup G'
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedAddCommGroup F''
inst✝⁵ : NormedAddCommGroup G''
inst✝⁴ : SeminormedRing R
inst✝³ : SeminormedAddGroup E'''
inst✝² : SeminormedRing R'
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : NormedDivisionRing 𝕜'
c c' c₁ c₂ : ℝ
f : α → E
g : α → F
k : α → G
f' : α → E'
g' : α → F'
k' : α → G'
f'' : α → E''
g'' : α → F''
k'' : α → G''
l l' : Filter α
x : α
⊢ x ∈ {x | (fun x => ‖0‖ ≤ 0 * ‖g x‖) x}
state_after: no goals
Theorem: finprod_mem_pair
File: Mathlib/Algebra/BigOperators/Finprod.lean

tactic: rw [finprod_mem_insert, finprod_mem_singleton]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f g : α → M
a b : α
s t : Set α
h : a ≠ b
⊢ ∏ᶠ (i : α) (_ : i ∈ {a, b}), f i = f a * f b
state_after: case h
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f g : α → M
a b : α
s t : Set α
h : a ≠ b
⊢ a ∉ {b}

case hs
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f g : α → M
a b : α
s t : Set α
h : a ≠ b
⊢ {b}.Finite

tactic: exacts [h, finite_singleton b]
state_before: case h
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f g : α → M
a b : α
s t : Set α
h : a ≠ b
⊢ a ∉ {b}

case hs
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f g : α → M
a b : α
s t : Set α
h : a ≠ b
⊢ {b}.Finite
state_after: no goals
Theorem: Finset.sdiff_singleton_eq_self
File: Mathlib/Data/Finset/Basic.lean

tactic: simp [ha]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s t u v : Finset α
a b : α
ha : a ∉ s
⊢ _root_.Disjoint {a} s
state_after: no goals
Theorem: ContinuousMap.Homotopy.apply_one_path
File: Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean

tactic: apply @eq_path_of_eq_image _ _ _ _ H.uliftMap _ _ _ _ _ ((Path.refl (ULift.up _)).prod p')
state_before: X Y : TopCat
f g : C(↑X, ↑Y)
H : f.Homotopy g
x₀ x₁ : ↑X
p : fromTop x₀ ⟶ fromTop x₁
p' : Path (fromTop x₀).as (fromTop x₁).as
⊢ (π.map g).map ⟦p'⟧ = hcast ⋯ ≫ (π.map H.uliftMap).map (prodToProdTopI (𝟙 (fromTop { down := 1 })) ⟦p'⟧) ≫ hcast ⋯
state_after: X Y : TopCat
f g : C(↑X, ↑Y)
H : f.Homotopy g
x₀ x₁ : ↑X
p : fromTop x₀ ⟶ fromTop x₁
p' : Path (fromTop x₀).as (fromTop x₁).as
⊢ ∀ (t : ↑I), g (p' t) = H.uliftMap (((Path.refl { down := 1 }).prod p') t)

tactic: erw [Path.prod_coe]
state_before: X Y : TopCat
f g : C(↑X, ↑Y)
H : f.Homotopy g
x₀ x₁ : ↑X
p : fromTop x₀ ⟶ fromTop x₁
p' : Path (fromTop x₀).as (fromTop x₁).as
⊢ ∀ (t : ↑I), g (p' t) = H.uliftMap (((Path.refl { down := 1 }).prod p') t)
state_after: X Y : TopCat
f g : C(↑X, ↑Y)
H : f.Homotopy g
x₀ x₁ : ↑X
p : fromTop x₀ ⟶ fromTop x₁
p' : Path (fromTop x₀).as (fromTop x₁).as
⊢ ∀ (t : ↑I), g (p' t) = H.uliftMap ((fun t => ((Path.refl { down := 1 }) t, p' t)) t)

tactic: simp_rw [ulift_apply]
state_before: X Y : TopCat
f g : C(↑X, ↑Y)
H : f.Homotopy g
x₀ x₁ : ↑X
p : fromTop x₀ ⟶ fromTop x₁
p' : Path (fromTop x₀).as (fromTop x₁).as
⊢ ∀ (t : ↑I), g (p' t) = H.uliftMap ((fun t => ((Path.refl { down := 1 }) t, p' t)) t)
state_after: X Y : TopCat
f g : C(↑X, ↑Y)
H : f.Homotopy g
x₀ x₁ : ↑X
p : fromTop x₀ ⟶ fromTop x₁
p' : Path (fromTop x₀).as (fromTop x₁).as
⊢ ∀ (t : ↑I), g (p' t) = H (((Path.refl { down := 1 }) t).down, p' t)

tactic: simp
state_before: X Y : TopCat
f g : C(↑X, ↑Y)
H : f.Homotopy g
x₀ x₁ : ↑X
p : fromTop x₀ ⟶ fromTop x₁
p' : Path (fromTop x₀).as (fromTop x₁).as
⊢ ∀ (t : ↑I), g (p' t) = H (((Path.refl { down := 1 }) t).down, p' t)
state_after: no goals
Theorem: Filter.eventuallyLE_antisymm_iff
File: Mathlib/Order/Filter/Basic.lean

tactic: simp only [EventuallyEq, EventuallyLE, le_antisymm_iff, eventually_and]
state_before: α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
l✝ : Filter α
inst✝ : PartialOrder β
l : Filter α
f g : α → β
⊢ f =ᶠ[l] g ↔ f ≤ᶠ[l] g ∧ g ≤ᶠ[l] f
state_after: no goals
Theorem: Function.support_const_smul_of_ne_zero
File: Mathlib/Algebra/Module/Basic.lean

tactic: simp only [hc, mem_support, Pi.smul_apply, Ne, smul_eq_zero, false_or_iff]
state_before: α : Type u_1
R : Type u_2
M : Type u_3
M₂ : Type u_4
inst✝³ : Zero R
inst✝² : Zero M
inst✝¹ : SMulWithZero R M
inst✝ : NoZeroSMulDivisors R M
c : R
g : α → M
hc : c ≠ 0
x : α
⊢ x ∈ support (c • g) ↔ x ∈ support g
state_after: no goals
Theorem: Fintype.univ_Prop
File: Mathlib/Data/Fintype/Basic.lean

tactic: simp
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
⊢ univ.val = {True, False}.val
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
⊢ univ.val = True ::ₘ {False}

tactic: rfl
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
⊢ univ.val = True ::ₘ {False}
state_after: no goals
Theorem: Submodule.map_comap_subtype
File: Mathlib/Algebra/Module/Submodule/Map.lean

tactic: rintro ⟨⟨_, h₁⟩, h₂, rfl⟩
state_before: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : Semiring R₃
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : Module R M
inst✝⁴ : Module R₂ M₂
inst✝³ : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
p p' : Submodule R M
q q' : Submodule R₂ M₂
x✝ : M
F : Type u_9
inst✝¹ : FunLike F M M₂
inst✝ : SemilinearMapClass F σ₁₂ M M₂
x : M
⊢ x ∈ map p.subtype (comap p.subtype p') → x ∈ p ⊓ p'
state_after: case intro.mk.intro
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : Semiring R₃
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : Module R M
inst✝⁴ : Module R₂ M₂
inst✝³ : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
p p' : Submodule R M
q q' : Submodule R₂ M₂
x : M
F : Type u_9
inst✝¹ : FunLike F M M₂
inst✝ : SemilinearMapClass F σ₁₂ M M₂
val✝ : M
h₁ : val✝ ∈ p
h₂ : ⟨val✝, h₁⟩ ∈ ↑(comap p.subtype p')
⊢ p.subtype ⟨val✝, h₁⟩ ∈ p ⊓ p'

tactic: exact ⟨h₁, h₂⟩
state_before: case intro.mk.intro
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : Semiring R₃
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : Module R M
inst✝⁴ : Module R₂ M₂
inst✝³ : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
p p' : Submodule R M
q q' : Submodule R₂ M₂
x : M
F : Type u_9
inst✝¹ : FunLike F M M₂
inst✝ : SemilinearMapClass F σ₁₂ M M₂
val✝ : M
h₁ : val✝ ∈ p
h₂ : ⟨val✝, h₁⟩ ∈ ↑(comap p.subtype p')
⊢ p.subtype ⟨val✝, h₁⟩ ∈ p ⊓ p'
state_after: no goals
Theorem: mul_tsub_one
File: Mathlib/Algebra/Order/Ring/Canonical.lean

tactic: rw [mul_tsub, mul_one]
state_before: α : Type u
β : Type u_1
inst✝⁴ : CanonicallyOrderedCommSemiring α
a✝ b✝ c : α
inst✝³ : Sub α
inst✝² : OrderedSub α
inst✝¹ : IsTotal α fun x x_1 => x ≤ x_1
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
a b : α
⊢ a * (b - 1) = a * b - a
state_after: no goals
Theorem: Function.Antiperiodic.int_odd_mul_antiperiodic
File: Mathlib/Algebra/Periodic.lean

tactic: rw [← add_assoc, h, h.int_even_mul_periodic]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
f g : α → β
c c₁ c₂ x✝ : α
inst✝¹ : Ring α
inst✝ : InvolutiveNeg β
h : Antiperiodic f c
n : ℤ
x : α
⊢ f (x + (↑n * (2 * c) + c)) = -f x
state_after: no goals
Theorem: BoundedContinuousFunction.toLp_denseRange
File: Mathlib/MeasureTheory/Function/ContinuousMapDense.lean

tactic: haveI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
state_before: α : Type u_1
inst✝¹⁰ : MeasurableSpace α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : T4Space α
inst✝⁷ : BorelSpace α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝⁵ : SecondCountableTopologyEither α E
_i : Fact (1 ≤ p)
hp : p ≠ ⊤
𝕜 : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedAlgebra ℝ 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : μ.WeaklyRegular
inst✝ : IsFiniteMeasure μ
⊢ DenseRange ⇑(toLp p μ 𝕜)
state_after: α : Type u_1
inst✝¹⁰ : MeasurableSpace α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : T4Space α
inst✝⁷ : BorelSpace α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝⁵ : SecondCountableTopologyEither α E
_i : Fact (1 ≤ p)
hp : p ≠ ⊤
𝕜 : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedAlgebra ℝ 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : μ.WeaklyRegular
inst✝ : IsFiniteMeasure μ
this : NormedSpace ℝ E
⊢ DenseRange ⇑(toLp p μ 𝕜)

tactic: simpa only [← range_toLp p μ (𝕜 := 𝕜)]
  using MeasureTheory.Lp.boundedContinuousFunction_dense E μ hp
state_before: α : Type u_1
inst✝¹⁰ : MeasurableSpace α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : T4Space α
inst✝⁷ : BorelSpace α
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝⁵ : SecondCountableTopologyEither α E
_i : Fact (1 ≤ p)
hp : p ≠ ⊤
𝕜 : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedAlgebra ℝ 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : μ.WeaklyRegular
inst✝ : IsFiniteMeasure μ
this : NormedSpace ℝ E
⊢ DenseRange ⇑(toLp p μ 𝕜)
state_after: no goals
Theorem: Real.aemeasurable_of_aemeasurable_exp
File: Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean

tactic: have : f = fun x ↦ log (exp (f x)) := by ext; rw [log_exp]
state_before: α : Type u_1
x✝ : MeasurableSpace α
f : α → ℝ
μ : MeasureTheory.Measure α
hf : AEMeasurable (fun x => rexp (f x)) μ
⊢ AEMeasurable f μ
state_after: α : Type u_1
x✝ : MeasurableSpace α
f : α → ℝ
μ : MeasureTheory.Measure α
hf : AEMeasurable (fun x => rexp (f x)) μ
this : f = fun x => log (rexp (f x))
⊢ AEMeasurable f μ

tactic: rw [this]
state_before: α : Type u_1
x✝ : MeasurableSpace α
f : α → ℝ
μ : MeasureTheory.Measure α
hf : AEMeasurable (fun x => rexp (f x)) μ
this : f = fun x => log (rexp (f x))
⊢ AEMeasurable f μ
state_after: α : Type u_1
x✝ : MeasurableSpace α
f : α → ℝ
μ : MeasureTheory.Measure α
hf : AEMeasurable (fun x => rexp (f x)) μ
this : f = fun x => log (rexp (f x))
⊢ AEMeasurable (fun x => log (rexp (f x))) μ

tactic: exact measurable_log.comp_aemeasurable hf
state_before: α : Type u_1
x✝ : MeasurableSpace α
f : α → ℝ
μ : MeasureTheory.Measure α
hf : AEMeasurable (fun x => rexp (f x)) μ
this : f = fun x => log (rexp (f x))
⊢ AEMeasurable (fun x => log (rexp (f x))) μ
state_after: no goals

tactic: ext
state_before: α : Type u_1
x✝ : MeasurableSpace α
f : α → ℝ
μ : MeasureTheory.Measure α
hf : AEMeasurable (fun x => rexp (f x)) μ
⊢ f = fun x => log (rexp (f x))
state_after: case h
α : Type u_1
x✝¹ : MeasurableSpace α
f : α → ℝ
μ : MeasureTheory.Measure α
hf : AEMeasurable (fun x => rexp (f x)) μ
x✝ : α
⊢ f x✝ = log (rexp (f x✝))

tactic: rw [log_exp]
state_before: case h
α : Type u_1
x✝¹ : MeasurableSpace α
f : α → ℝ
μ : MeasureTheory.Measure α
hf : AEMeasurable (fun x => rexp (f x)) μ
x✝ : α
⊢ f x✝ = log (rexp (f x✝))
state_after: no goals
Theorem: Int.coprime_of_sq_sum'
File: Mathlib/NumberTheory/FLT/Four.lean

tactic: apply IsCoprime.mul_right (Int.coprime_of_sq_sum (isCoprime_comm.mp h))
state_before: r s : ℤ
h : IsCoprime r s
⊢ IsCoprime (r ^ 2 + s ^ 2) (r * s)
state_after: r s : ℤ
h : IsCoprime r s
⊢ IsCoprime (r ^ 2 + s ^ 2) s

tactic: rw [add_comm]
state_before: r s : ℤ
h : IsCoprime r s
⊢ IsCoprime (r ^ 2 + s ^ 2) s
state_after: r s : ℤ
h : IsCoprime r s
⊢ IsCoprime (s ^ 2 + r ^ 2) s

tactic: apply Int.coprime_of_sq_sum h
state_before: r s : ℤ
h : IsCoprime r s
⊢ IsCoprime (s ^ 2 + r ^ 2) s
state_after: no goals
Theorem: CharP.ringChar_ne_one
File: Mathlib/Algebra/CharP/Defs.lean

tactic: intro h
state_before: R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : Nontrivial R
⊢ ringChar R ≠ 1
state_after: R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : Nontrivial R
h : ringChar R = 1
⊢ False

tactic: apply zero_ne_one' R
state_before: R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : Nontrivial R
h : ringChar R = 1
⊢ False
state_after: R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : Nontrivial R
h : ringChar R = 1
⊢ 0 = 1

tactic: symm
state_before: R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : Nontrivial R
h : ringChar R = 1
⊢ 0 = 1
state_after: R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : Nontrivial R
h : ringChar R = 1
⊢ 1 = 0

tactic: rw [← Nat.cast_one, ringChar.spec, h]
state_before: R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : Nontrivial R
h : ringChar R = 1
⊢ 1 = 0
state_after: no goals
Theorem: Part.union_get_eq
File: Mathlib/Data/Part.lean

tactic: simp [union_def]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : Union α
a b : Part α
hab : (a ∪ b).Dom
⊢ (a ∪ b).get hab = a.get ⋯ ∪ b.get ⋯
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : Union α
a b : Part α
hab : (a ∪ b).Dom
⊢ (a.bind fun y => map (fun x => y ∪ x) b).get ⋯ = a.get ⋯ ∪ b.get ⋯

tactic: aesop
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : Union α
a b : Part α
hab : (a ∪ b).Dom
⊢ (a.bind fun y => map (fun x => y ∪ x) b).get ⋯ = a.get ⋯ ∪ b.get ⋯
state_after: no goals
Theorem: Tactic.ReduceModChar.CharP.neg_mul_eq_sub_one_mul
File: Mathlib/Tactic/ReduceModChar.lean

tactic: rw [← pa, ← p.out, ← neg_one_mul]
state_before: α : Type u_1
inst✝ : Ring α
n : ℕ
inst : CharP α n
a b : α
na : ℕ
na' : α
p : IsNat ((↑n - 1) * a) na
pa : ↑na = na'
⊢ -(a * b) = na' * b
state_after: α : Type u_1
inst✝ : Ring α
n : ℕ
inst : CharP α n
a b : α
na : ℕ
na' : α
p : IsNat ((↑n - 1) * a) na
pa : ↑na = na'
⊢ -1 * (a * b) = (↑n - 1) * a * b

tactic: simp
state_before: α : Type u_1
inst✝ : Ring α
n : ℕ
inst : CharP α n
a b : α
na : ℕ
na' : α
p : IsNat ((↑n - 1) * a) na
pa : ↑na = na'
⊢ -1 * (a * b) = (↑n - 1) * a * b
state_after: no goals
Theorem: ENNReal.toReal_ofReal_mul
File: Mathlib/Data/ENNReal/Real.lean

tactic: rw [ENNReal.toReal_mul, ENNReal.toReal_ofReal h]
state_before: a✝ b c✝ d : ℝ≥0∞
r p q : ℝ≥0
c : ℝ
a : ℝ≥0∞
h : 0 ≤ c
⊢ (ENNReal.ofReal c * a).toReal = c * a.toReal
state_after: no goals
Theorem: FiniteField.pow_card_sub_one_eq_one
File: Mathlib/FieldTheory/Finite/Basic.lean

tactic: calc
  a ^ (Fintype.card K - 1) = (Units.mk0 a ha ^ (Fintype.card K - 1) : Kˣ).1 := by
    rw [Units.val_pow_eq_pow_val, Units.val_mk0]
  _ = 1 := by
    classical
      rw [← Fintype.card_units, pow_card_eq_one]
      rfl
state_before: K : Type u_1
R : Type u_2
inst✝¹ : GroupWithZero K
inst✝ : Fintype K
a : K
ha : a ≠ 0
⊢ a ^ (q - 1) = 1
state_after: no goals

tactic: rw [Units.val_pow_eq_pow_val, Units.val_mk0]
state_before: K : Type u_1
R : Type u_2
inst✝¹ : GroupWithZero K
inst✝ : Fintype K
a : K
ha : a ≠ 0
⊢ a ^ (q - 1) = ↑(Units.mk0 a ha ^ (q - 1))
state_after: no goals

tactic: classical
  rw [← Fintype.card_units, pow_card_eq_one]
  rfl
state_before: K : Type u_1
R : Type u_2
inst✝¹ : GroupWithZero K
inst✝ : Fintype K
a : K
ha : a ≠ 0
⊢ ↑(Units.mk0 a ha ^ (q - 1)) = 1
state_after: no goals

tactic: rw [← Fintype.card_units, pow_card_eq_one]
state_before: K : Type u_1
R : Type u_2
inst✝¹ : GroupWithZero K
inst✝ : Fintype K
a : K
ha : a ≠ 0
⊢ ↑(Units.mk0 a ha ^ (q - 1)) = 1
state_after: K : Type u_1
R : Type u_2
inst✝¹ : GroupWithZero K
inst✝ : Fintype K
a : K
ha : a ≠ 0
⊢ ↑1 = 1

tactic: rfl
state_before: K : Type u_1
R : Type u_2
inst✝¹ : GroupWithZero K
inst✝ : Fintype K
a : K
ha : a ≠ 0
⊢ ↑1 = 1
state_after: no goals
Theorem: MultilinearMap.map_coord_zero
File: Mathlib/LinearAlgebra/Multilinear/Basic.lean

tactic: classical
  have : (0 : R) • (0 : M₁ i) = 0 := by simp
  rw [← update_eq_self i m, h, ← this, f.map_smul, zero_smul R (M := M₂)]
state_before: R : Type uR
S : Type uS
ι : Type uι
n : ℕ
M : Fin n.succ → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝¹⁰ : Semiring R
inst✝⁹ : (i : Fin n.succ) → AddCommMonoid (M i)
inst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : AddCommMonoid M'
inst✝⁴ : (i : Fin n.succ) → Module R (M i)
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
inst✝¹ : Module R M₃
inst✝ : Module R M'
f f' : MultilinearMap R M₁ M₂
m : (i : ι) → M₁ i
i : ι
h : m i = 0
⊢ f m = 0
state_after: no goals

tactic: have : (0 : R) • (0 : M₁ i) = 0 := by simp
state_before: R : Type uR
S : Type uS
ι : Type uι
n : ℕ
M : Fin n.succ → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝¹⁰ : Semiring R
inst✝⁹ : (i : Fin n.succ) → AddCommMonoid (M i)
inst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : AddCommMonoid M'
inst✝⁴ : (i : Fin n.succ) → Module R (M i)
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
inst✝¹ : Module R M₃
inst✝ : Module R M'
f f' : MultilinearMap R M₁ M₂
m : (i : ι) → M₁ i
i : ι
h : m i = 0
⊢ f m = 0
state_after: R : Type uR
S : Type uS
ι : Type uι
n : ℕ
M : Fin n.succ → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝¹⁰ : Semiring R
inst✝⁹ : (i : Fin n.succ) → AddCommMonoid (M i)
inst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : AddCommMonoid M'
inst✝⁴ : (i : Fin n.succ) → Module R (M i)
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
inst✝¹ : Module R M₃
inst✝ : Module R M'
f f' : MultilinearMap R M₁ M₂
m : (i : ι) → M₁ i
i : ι
h : m i = 0
this : 0 • 0 = 0
⊢ f m = 0

tactic: rw [← update_eq_self i m, h, ← this, f.map_smul, zero_smul R (M := M₂)]
state_before: R : Type uR
S : Type uS
ι : Type uι
n : ℕ
M : Fin n.succ → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝¹⁰ : Semiring R
inst✝⁹ : (i : Fin n.succ) → AddCommMonoid (M i)
inst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : AddCommMonoid M'
inst✝⁴ : (i : Fin n.succ) → Module R (M i)
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
inst✝¹ : Module R M₃
inst✝ : Module R M'
f f' : MultilinearMap R M₁ M₂
m : (i : ι) → M₁ i
i : ι
h : m i = 0
this : 0 • 0 = 0
⊢ f m = 0
state_after: no goals

tactic: simp
state_before: R : Type uR
S : Type uS
ι : Type uι
n : ℕ
M : Fin n.succ → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝¹⁰ : Semiring R
inst✝⁹ : (i : Fin n.succ) → AddCommMonoid (M i)
inst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : AddCommMonoid M'
inst✝⁴ : (i : Fin n.succ) → Module R (M i)
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
inst✝¹ : Module R M₃
inst✝ : Module R M'
f f' : MultilinearMap R M₁ M₂
m : (i : ι) → M₁ i
i : ι
h : m i = 0
⊢ 0 • 0 = 0
state_after: no goals
Theorem: Besicovitch.TauPackage.monotone_iUnionUpTo
File: Mathlib/MeasureTheory/Covering/Besicovitch.lean

tactic: intro i j hij
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
⊢ Monotone p.iUnionUpTo
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i j : Ordinal.{u}
hij : i ≤ j
⊢ p.iUnionUpTo i ≤ p.iUnionUpTo j

tactic: simp only [iUnionUpTo]
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i j : Ordinal.{u}
hij : i ≤ j
⊢ p.iUnionUpTo i ≤ p.iUnionUpTo j
state_after: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i j : Ordinal.{u}
hij : i ≤ j
⊢ ⋃ j, ball (p.c (p.index ↑j)) (p.r (p.index ↑j)) ≤ ⋃ j_1, ball (p.c (p.index ↑j_1)) (p.r (p.index ↑j_1))

tactic: exact iUnion_mono' fun r => ⟨⟨r, r.2.trans_le hij⟩, Subset.rfl⟩
state_before: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i j : Ordinal.{u}
hij : i ≤ j
⊢ ⋃ j, ball (p.c (p.index ↑j)) (p.r (p.index ↑j)) ≤ ⋃ j_1, ball (p.c (p.index ↑j_1)) (p.r (p.index ↑j_1))
state_after: no goals
Theorem: Cardinal.lift_eq_ofENat
File: Mathlib/SetTheory/Cardinal/ENat.lean

tactic: rw [← lift_ofENat.{u, v}, lift_inj]
state_before: x : Cardinal.{v}
m : ℕ∞
⊢ lift.{u, v} x = ↑m ↔ x = ↑m
state_after: no goals
Theorem: List.inits_append
File: Mathlib/Data/List/Infix.lean

tactic: simp
state_before: α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m n : ℕ
⊢ ([] ++ []).inits = [].inits ++ map (fun l => [] ++ l) [].inits.tail
state_after: no goals