ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: measurable_measure_prod_mk_left
File: Mathlib/MeasureTheory/Constructions/Prod/Basic.lean

Proof steps:
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
⊢ Measurable fun x => ν (Prod.mk x ⁻¹' s)
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
⊢ Measurable fun x => (sum (sFiniteSeq ν)) (Prod.mk x ⁻¹' s)
tactic: rw [← sum_sFiniteSeq ν]
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
⊢ Measurable fun x => (sum (sFiniteSeq ν)) (Prod.mk x ⁻¹' s)
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
⊢ Measurable fun x => ∑' (i : ℕ), (sFiniteSeq ν i) (Prod.mk x ⁻¹' s)
tactic: simp_rw [Measure.sum_apply_of_countable]
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
⊢ Measurable fun x => ∑' (i : ℕ), (sFiniteSeq ν i) (Prod.mk x ⁻¹' s)
state_after: no goals
tactic: exact Measurable.ennreal_tsum (fun i ↦ measurable_measure_prod_mk_left_finite hs)
Theorem: Finset.memberSubfamily_union
File: Mathlib/Combinatorics/SetFamily/Compression/Down.lean

Proof steps:
state_before: α : Type u_1
inst✝ : DecidableEq α
𝒜✝ ℬ✝ : Finset (Finset α)
s : Finset α
a✝ a : α
𝒜 ℬ : Finset (Finset α)
⊢ memberSubfamily a (𝒜 ∪ ℬ) = memberSubfamily a 𝒜 ∪ memberSubfamily a ℬ
state_after: no goals
tactic: simp_rw [memberSubfamily, filter_union, image_union]
Theorem: strictAnti_of_hasDerivAt_neg
File: Mathlib/Analysis/Calculus/MeanValue.lean

Proof steps:
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
hf : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : ∀ (x : ℝ), f' x < 0
x : ℝ
⊢ deriv f x < 0
state_after: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
hf : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : ∀ (x : ℝ), f' x < 0
x : ℝ
⊢ f' x < 0
tactic: rw [(hf _).deriv]
state_before: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
hf : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : ∀ (x : ℝ), f' x < 0
x : ℝ
⊢ f' x < 0
state_after: no goals
tactic: exact hf' _
Theorem: Finset.sum_subset_zero_on_sdiff
File: Mathlib/Algebra/BigOperators/Group/Finset.lean

Proof steps:
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✝ : DecidableEq α
h : s₁ ⊆ s₂
hg : ∀ x ∈ s₂ \ s₁, g x = 1
hfg : ∀ x ∈ s₁, f x = g x
⊢ ∏ i ∈ s₁, f i = ∏ i ∈ s₂, g i
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✝ : DecidableEq α
h : s₁ ⊆ s₂
hg : ∀ x ∈ s₂ \ s₁, g x = 1
hfg : ∀ x ∈ s₁, f x = g x
⊢ ∏ i ∈ s₁, f i = ∏ x ∈ s₁, g x
tactic: rw [← prod_sdiff h, prod_eq_one hg, one_mul]
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✝ : DecidableEq α
h : s₁ ⊆ s₂
hg : ∀ x ∈ s₂ \ s₁, g x = 1
hfg : ∀ x ∈ s₁, f x = g x
⊢ ∏ i ∈ s₁, f i = ∏ x ∈ s₁, g x
state_after: no goals
tactic: exact prod_congr rfl hfg
Theorem: Multiset.map_toEnumFinset_fst
File: Mathlib/Data/Multiset/Fintype.lean

Proof steps:
state_before: α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ map Prod.fst m.toEnumFinset.val = m
state_after: case a
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
a : α
⊢ count a (map Prod.fst m.toEnumFinset.val) = count a m
tactic: ext a
state_before: case a
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
a : α
⊢ count a (map Prod.fst m.toEnumFinset.val) = count a m
state_after: no goals
tactic: simp [count_map, ← Finset.filter_val, eq_comm (a := a)]
Theorem: MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable₀
File: Mathlib/MeasureTheory/Measure/WithDensity.lean

Proof steps:
state_before: α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤
g : α → ℝ≥0∞
⊢ ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ
state_after: α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
⊢ ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ
tactic: let f' := hf.mk f
state_before: α : Type u_1
m0 : MeasurableSpace α
μ✝ μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
⊢ ∫⁻ (a : α), g a ∂μ.withDensity f = ∫⁻ (a : α), (f * g) a ∂μ
state_after: no goals
tactic: calc
    ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, g a ∂μ.withDensity f' := by
      rw [withDensity_congr_ae hf.ae_eq_mk]
    _ = ∫⁻ a, (f' * g) a ∂μ := by
      apply lintegral_withDensity_eq_lintegral_mul_non_measurable _ hf.measurable_mk
      filter_upwards [h'f, hf.ae_eq_mk]
      intro x hx h'x
      rwa [← h'x]
    _ = ∫⁻ a, (f * g) a ∂μ := by
      apply lintegral_congr_ae
      filter_upwards [hf.ae_eq_mk]
      intro x hx
      simp only [hx, Pi.mul_apply]
Theorem: Int.floor_add_nat
File: Mathlib/Algebra/Order/Floor.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
z : ℤ
a✝ a : α
n : ℕ
⊢ ⌊a + ↑n⌋ = ⌊a⌋ + ↑n
state_after: no goals
tactic: rw [← Int.cast_natCast, floor_add_int]
Theorem: AddSubgroup.subsingleton_iff
File: Mathlib/Algebra/Group/Subgroup/Basic.lean

Proof steps:
state_before: G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
H K : Subgroup G
h : Subsingleton G
x y : Subgroup G
i : G
⊢ 1 ∈ x ↔ 1 ∈ y
state_after: no goals
tactic: simp [Subgroup.one_mem]
Theorem: inner_add_right
File: Mathlib/Analysis/InnerProductSpace/Basic.lean

Proof steps:
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : _root_.RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y z : E
⊢ ⟪x, y + z⟫_𝕜 = ⟪x, y⟫_𝕜 + ⟪x, z⟫_𝕜
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : _root_.RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y z : E
⊢ (starRingEnd 𝕜) ⟪y, x⟫_𝕜 + (starRingEnd 𝕜) ⟪z, x⟫_𝕜 = ⟪x, y⟫_𝕜 + ⟪x, z⟫_𝕜
tactic: rw [← inner_conj_symm, inner_add_left, RingHom.map_add]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : _root_.RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y z : E
⊢ (starRingEnd 𝕜) ⟪y, x⟫_𝕜 + (starRingEnd 𝕜) ⟪z, x⟫_𝕜 = ⟪x, y⟫_𝕜 + ⟪x, z⟫_𝕜
state_after: no goals
tactic: simp only [inner_conj_symm]
Theorem: Euclidean.closedBall_eq_image
File: Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean

Proof steps:
state_before: E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : T2Space E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
r : ℝ
⊢ closedBall x r = ⇑toEuclidean.symm '' Metric.closedBall (toEuclidean x) r
state_after: no goals
tactic: rw [toEuclidean.image_symm_eq_preimage, closedBall_eq_preimage]
Theorem: MeasureTheory.OuterMeasure.smul_dirac_apply
File: Mathlib/MeasureTheory/OuterMeasure/Operations.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
m : OuterMeasure α
a : ℝ≥0∞
b : α
s : Set α
⊢ (a • dirac b) s = s.indicator (fun x => a) b
state_after: no goals
tactic: simp only [smul_apply, smul_eq_mul, dirac_apply, ← indicator_mul_right _ fun _ => a, mul_one]
Theorem: Polynomial.evalEval_X
File: Mathlib/Algebra/Polynomial/Bivariate.lean

Proof steps:
state_before: R : Type u_1
S : Type u_2
inst✝ : Semiring R
x y : R
⊢ evalEval x y Y = y
state_after: no goals
tactic: rw [evalEval, eval_X, eval_C]
Theorem: SetTheory.PGame.add_moveRight_inl
File: Mathlib/SetTheory/Game/PGame.lean

Proof steps:
state_before: xl xr : Type u
x y : PGame
i : x.RightMoves
⊢ (x + y).moveRight (toRightMovesAdd (Sum.inl i)) = x.moveRight i + y
state_after: case mk
xl xr : Type u
y : PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
i : (mk α✝ β✝ a✝¹ a✝).RightMoves
⊢ (mk α✝ β✝ a✝¹ a✝ + y).moveRight (toRightMovesAdd (Sum.inl i)) = (mk α✝ β✝ a✝¹ a✝).moveRight i + y
tactic: cases x
state_before: case mk
xl xr : Type u
y : PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
i : (mk α✝ β✝ a✝¹ a✝).RightMoves
⊢ (mk α✝ β✝ a✝¹ a✝ + y).moveRight (toRightMovesAdd (Sum.inl i)) = (mk α✝ β✝ a✝¹ a✝).moveRight i + y
state_after: case mk.mk
xl xr : Type u
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
i : (mk α✝¹ β✝¹ a✝³ a✝²).RightMoves
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
⊢ (mk α✝¹ β✝¹ a✝³ a✝² + mk α✝ β✝ a✝¹ a✝).moveRight (toRightMovesAdd (Sum.inl i)) =
    (mk α✝¹ β✝¹ a✝³ a✝²).moveRight i + mk α✝ β✝ a✝¹ a✝
tactic: cases y
state_before: case mk.mk
xl xr : Type u
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
i : (mk α✝¹ β✝¹ a✝³ a✝²).RightMoves
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
⊢ (mk α✝¹ β✝¹ a✝³ a✝² + mk α✝ β✝ a✝¹ a✝).moveRight (toRightMovesAdd (Sum.inl i)) =
    (mk α✝¹ β✝¹ a✝³ a✝²).moveRight i + mk α✝ β✝ a✝¹ a✝
state_after: no goals
tactic: rfl
Theorem: Finset.compls_subset_iff
File: Mathlib/Data/Finset/Sups.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
inst✝² : DecidableEq α
inst✝¹ : DecidableEq β
inst✝ : BooleanAlgebra α
s s₁ s₂ t t₁ t₂ u v : Finset α
a b c : α
⊢ sᶜˢ ⊆ t ↔ s ⊆ tᶜˢ
state_after: no goals
tactic: rw [← compls_subset_compls, compls_compls]
Theorem: Finset.noncommSum_cons'
File: Mathlib/Data/Finset/NoncommProd.lean

Proof steps:
state_before: F : Type u_1
ι : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
f✝ : α → β → β
op : α → α → α
inst✝¹ : Monoid β
inst✝ : Monoid γ
s : Finset α
a : α
f : α → β
ha : a ∉ s
comm : (↑(cons a s ha)).Pairwise fun a b => Commute (f a) (f b)
⊢ (cons a s ha).noncommProd f comm = s.noncommProd f ⋯ * f a
state_after: no goals
tactic: simp_rw [noncommProd, Finset.cons_val, Multiset.map_cons, Multiset.noncommProd_cons']
Theorem: Orientation.two_zsmul_oangle_left_of_span_eq
File: Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean

Proof steps:
state_before: V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
h : Submodule.span ℝ {x} = Submodule.span ℝ {y}
⊢ 2 • o.oangle x z = 2 • o.oangle y z
state_after: V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
h : ∃ z, z • x = y
⊢ 2 • o.oangle x z = 2 • o.oangle y z
tactic: rw [Submodule.span_singleton_eq_span_singleton] at h
state_before: V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
h : ∃ z, z • x = y
⊢ 2 • o.oangle x z = 2 • o.oangle y z
state_after: case intro
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x z : V
r : ℝˣ
⊢ 2 • o.oangle x z = 2 • o.oangle (r • x) z
tactic: rcases h with ⟨r, rfl⟩
state_before: case intro
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x z : V
r : ℝˣ
⊢ 2 • o.oangle x z = 2 • o.oangle (r • x) z
state_after: no goals
tactic: exact (o.two_zsmul_oangle_smul_left_of_ne_zero _ _ (Units.ne_zero _)).symm
Theorem: Polynomial.card_support_le_one_iff_monomial
File: Mathlib/Algebra/Polynomial/Monomial.lean

Proof steps:
state_before: R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
p q r f : R[X]
⊢ f.support.card ≤ 1 ↔ ∃ n a, f = (monomial n) a
state_after: case mp
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
p q r f : R[X]
⊢ f.support.card ≤ 1 → ∃ n a, f = (monomial n) a

case mpr
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
p q r f : R[X]
⊢ (∃ n a, f = (monomial n) a) → f.support.card ≤ 1
tactic: constructor
Theorem: Matrix.charmatrix_reindex
File: Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean

Proof steps:
state_before: R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
M₁₁ : Matrix m m R
M₁₂ : Matrix m n R
M₂₁ : Matrix n m R
M₂₂ M : Matrix n n R
i j : n
e : n ≃ m
⊢ ((reindex e e) M).charmatrix = (reindex e e) M.charmatrix
state_after: case a.a
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
M₁₁ : Matrix m m R
M₁₂ : Matrix m n R
M₂₁ : Matrix n m R
M₂₂ M : Matrix n n R
i✝ j✝ : n
e : n ≃ m
i j : m
x : ℕ
⊢ (((reindex e e) M).charmatrix i j).coeff x = ((reindex e e) M.charmatrix i j).coeff x
tactic: ext i j x
state_before: case a.a
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
M₁₁ : Matrix m m R
M₁₂ : Matrix m n R
M₂₁ : Matrix n m R
M₂₂ M : Matrix n n R
i✝ j✝ : n
e : n ≃ m
i j : m
x : ℕ
⊢ (((reindex e e) M).charmatrix i j).coeff x = ((reindex e e) M.charmatrix i j).coeff x
state_after: case pos
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
M₁₁ : Matrix m m R
M₁₂ : Matrix m n R
M₂₁ : Matrix n m R
M₂₂ M : Matrix n n R
i✝ j✝ : n
e : n ≃ m
i j : m
x : ℕ
h : i = j
⊢ (((reindex e e) M).charmatrix i j).coeff x = ((reindex e e) M.charmatrix i j).coeff x

case neg
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
M₁₁ : Matrix m m R
M₁₂ : Matrix m n R
M₂₁ : Matrix n m R
M₂₂ M : Matrix n n R
i✝ j✝ : n
e : n ≃ m
i j : m
x : ℕ
h : ¬i = j
⊢ (((reindex e e) M).charmatrix i j).coeff x = ((reindex e e) M.charmatrix i j).coeff x
tactic: by_cases h : i = j
state_before: case pos
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
M₁₁ : Matrix m m R
M₁₂ : Matrix m n R
M₂₁ : Matrix n m R
M₂₂ M : Matrix n n R
i✝ j✝ : n
e : n ≃ m
i j : m
x : ℕ
h : i = j
⊢ (((reindex e e) M).charmatrix i j).coeff x = ((reindex e e) M.charmatrix i j).coeff x

case neg
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
M₁₁ : Matrix m m R
M₁₂ : Matrix m n R
M₂₁ : Matrix n m R
M₂₂ M : Matrix n n R
i✝ j✝ : n
e : n ≃ m
i j : m
x : ℕ
h : ¬i = j
⊢ (((reindex e e) M).charmatrix i j).coeff x = ((reindex e e) M.charmatrix i j).coeff x
state_after: no goals
tactic: all_goals simp [h]
Theorem: CategoryTheory.Presheaf.isSheaf_coherent_of_projective_of_comp
File: Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean

Proof steps:
state_before: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{?u.63975, u_2} D
F✝ : C ⥤ D
A : Type u₃
inst✝⁵ : Category.{v₃, u₃} A
F : Cᵒᵖ ⥤ A
B : Type u₄
inst✝⁴ : Category.{v₄, u₄} B
s : A ⥤ B
inst✝³ : Preregular C
inst✝² : FinitaryExtensive C
inst✝¹ : ∀ (X : C), Projective X
inst✝ : ReflectsFiniteProducts s
hF : IsSheaf (coherentTopology C) (F ⋙ s)
⊢ IsSheaf (coherentTopology C) F
state_after: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{?u.63975, u_2} D
F✝ : C ⥤ D
A : Type u₃
inst✝⁵ : Category.{v₃, u₃} A
F : Cᵒᵖ ⥤ A
B : Type u₄
inst✝⁴ : Category.{v₄, u₄} B
s : A ⥤ B
inst✝³ : Preregular C
inst✝² : FinitaryExtensive C
inst✝¹ : ∀ (X : C), Projective X
inst✝ : ReflectsFiniteProducts s
hF : Nonempty (PreservesFiniteProducts (F ⋙ s))
⊢ Nonempty (PreservesFiniteProducts F)
tactic: rw [isSheaf_iff_preservesFiniteProducts_of_projective] at hF ⊢
state_before: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{?u.63975, u_2} D
F✝ : C ⥤ D
A : Type u₃
inst✝⁵ : Category.{v₃, u₃} A
F : Cᵒᵖ ⥤ A
B : Type u₄
inst✝⁴ : Category.{v₄, u₄} B
s : A ⥤ B
inst✝³ : Preregular C
inst✝² : FinitaryExtensive C
inst✝¹ : ∀ (X : C), Projective X
inst✝ : ReflectsFiniteProducts s
hF : Nonempty (PreservesFiniteProducts (F ⋙ s))
⊢ Nonempty (PreservesFiniteProducts F)
state_after: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{?u.63975, u_2} D
F✝ : C ⥤ D
A : Type u₃
inst✝⁵ : Category.{v₃, u₃} A
F : Cᵒᵖ ⥤ A
B : Type u₄
inst✝⁴ : Category.{v₄, u₄} B
s : A ⥤ B
inst✝³ : Preregular C
inst✝² : FinitaryExtensive C
inst✝¹ : ∀ (X : C), Projective X
inst✝ : ReflectsFiniteProducts s
hF : Nonempty (PreservesFiniteProducts (F ⋙ s))
J : Type
x✝ : Fintype J
K : Discrete J ⥤ Cᵒᵖ
c : Cone K
hc : IsLimit c
⊢ IsLimit (F.mapCone c)
tactic: refine ⟨⟨fun J _ ↦ ⟨fun {K} ↦ ⟨fun {c} hc ↦ ?_⟩⟩⟩⟩
state_before: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{?u.63975, u_2} D
F✝ : C ⥤ D
A : Type u₃
inst✝⁵ : Category.{v₃, u₃} A
F : Cᵒᵖ ⥤ A
B : Type u₄
inst✝⁴ : Category.{v₄, u₄} B
s : A ⥤ B
inst✝³ : Preregular C
inst✝² : FinitaryExtensive C
inst✝¹ : ∀ (X : C), Projective X
inst✝ : ReflectsFiniteProducts s
hF : Nonempty (PreservesFiniteProducts (F ⋙ s))
J : Type
x✝ : Fintype J
K : Discrete J ⥤ Cᵒᵖ
c : Cone K
hc : IsLimit c
⊢ IsLimit (F.mapCone c)
state_after: no goals
tactic: exact isLimitOfReflects s ((hF.some.1 J).1.1 hc)
Theorem: suffixLevenshtein_cons₁_aux
File: Mathlib/Data/List/EditDistance/Defs.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
inst✝¹ : AddZeroClass δ
inst✝ : Min δ
C : Cost α β δ
x y : { r // 0 < r.length }
w₀ : x.val[0] = y.val[0]
w : x.val.tail = y.val.tail
⊢ x = y
state_after: no goals
tactic: match x, y with
  | ⟨hx :: tx, _⟩, ⟨hy :: ty, _⟩ => simp_all
Theorem: CategoryTheory.ShortComplex.opcycles_ext_iff
File: Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean

Proof steps:
state_before: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
inst✝ : S.HasRightHomology
A : C
f₁ f₂ : S.opcycles ⟶ A
⊢ f₁ = f₂ ↔ S.pOpcycles ≫ f₁ = S.pOpcycles ≫ f₂
state_after: no goals
tactic: rw [cancel_epi]
Theorem: vectorSpan_pair
File: Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean

Proof steps:
state_before: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p₁ p₂ : P
⊢ vectorSpan k {p₁, p₂} = Submodule.span k {p₁ -ᵥ p₂}
state_after: no goals
tactic: simp_rw [vectorSpan_eq_span_vsub_set_left k (mem_insert p₁ _), image_pair, vsub_self,
    Submodule.span_insert_zero]
Theorem: CategoryTheory.Subobject.factors_of_le
File: Mathlib/CategoryTheory/Subobject/FactorThru.lean

Proof steps:
state_before: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y✝ Z✝ : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
Y Z : C
P Q : Subobject Y
f : Z ⟶ Y
h : P ≤ Q
⊢ P.Factors f → Q.Factors f
state_after: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y✝ Z✝ : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
Y Z : C
P Q : Subobject Y
f : Z ⟶ Y
h : P ≤ Q
⊢ (representative.obj P).Factors f → (representative.obj Q).Factors f
tactic: simp only [factors_iff]
state_before: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y✝ Z✝ : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
Y Z : C
P Q : Subobject Y
f : Z ⟶ Y
h : P ≤ Q
⊢ (representative.obj P).Factors f → (representative.obj Q).Factors f
state_after: no goals
tactic: exact fun ⟨u, hu⟩ => ⟨u ≫ ofLE _ _ h, by simp [← hu]⟩
Theorem: MeasureTheory.SimpleFunc.restrict_univ
File: Mathlib/MeasureTheory/Function/SimpleFunc.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : MeasurableSpace α
K : Type u_5
inst✝ : Zero β
f : α →ₛ β
⊢ f.restrict univ = f
state_after: no goals
tactic: simp [restrict]
Theorem: closedBall_zero_sub_singleton
File: Mathlib/Analysis/Normed/Group/Pointwise.lean

Proof steps:
state_before: E : Type u_1
inst✝ : SeminormedCommGroup E
ε δ : ℝ
s t : Set E
x y : E
⊢ closedBall 1 δ / {x} = closedBall x⁻¹ δ
state_after: no goals
tactic: simp
Theorem: Finset.prod_eq_prod_Ico_succ_bot
File: Mathlib/Algebra/BigOperators/Intervals.lean

Proof steps:
state_before: α : Type u_1
M : Type u_2
inst✝ : CommMonoid M
s₂ s₁ s : Finset α
a✝ : α
g f✝ : α → M
a b : ℕ
hab : a < b
f : ℕ → M
⊢ ∏ k ∈ Ico a b, f k = f a * ∏ k ∈ Ico (a + 1) b, f k
state_after: α : Type u_1
M : Type u_2
inst✝ : CommMonoid M
s₂ s₁ s : Finset α
a✝ : α
g f✝ : α → M
a b : ℕ
hab : a < b
f : ℕ → M
ha : a ∉ Ico (a + 1) b
⊢ ∏ k ∈ Ico a b, f k = f a * ∏ k ∈ Ico (a + 1) b, f k
tactic: have ha : a ∉ Ico (a + 1) b := by simp
state_before: α : Type u_1
M : Type u_2
inst✝ : CommMonoid M
s₂ s₁ s : Finset α
a✝ : α
g f✝ : α → M
a b : ℕ
hab : a < b
f : ℕ → M
ha : a ∉ Ico (a + 1) b
⊢ ∏ k ∈ Ico a b, f k = f a * ∏ k ∈ Ico (a + 1) b, f k
state_after: no goals
tactic: rw [← prod_insert ha, Nat.Ico_insert_succ_left hab]