ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: Stream'.zip_eq
File: Mathlib/Data/Stream/Init.lean

Proof steps:
state_before: α : Type u
β : Type v
δ : Type w
f : α → β → δ
s₁ : Stream' α
s₂ : Stream' β
⊢ zip f s₁ s₂ = f s₁.head s₂.head :: zip f s₁.tail s₂.tail
state_after: α : Type u
β : Type v
δ : Type w
f : α → β → δ
s₁ : Stream' α
s₂ : Stream' β
⊢ (zip f s₁ s₂).head :: (zip f s₁ s₂).tail = f s₁.head s₂.head :: zip f s₁.tail s₂.tail
tactic: rw [← Stream'.eta (zip f s₁ s₂)]
state_before: α : Type u
β : Type v
δ : Type w
f : α → β → δ
s₁ : Stream' α
s₂ : Stream' β
⊢ (zip f s₁ s₂).head :: (zip f s₁ s₂).tail = f s₁.head s₂.head :: zip f s₁.tail s₂.tail
state_after: no goals
tactic: rfl
Theorem: EqualCharZero.pnatCast_eq_natCast
File: Mathlib/Algebra/CharP/MixedCharZero.lean

Proof steps:
state_before: R : Type u_1
inst✝¹ : CommRing R
inst✝ : Fact (∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I))
n : ℕ+
⊢ ↑↑n = ↑↑n
state_after: R : Type u_1
inst✝¹ : CommRing R
inst✝ : Fact (∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I))
n : ℕ+
⊢ ↑⋯.unit = ↑↑n
tactic: change ((PNat.isUnit_natCast (R := R) n).unit : R) = ↑n
state_before: R : Type u_1
inst✝¹ : CommRing R
inst✝ : Fact (∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I))
n : ℕ+
⊢ ↑⋯.unit = ↑↑n
state_after: no goals
tactic: simp only [IsUnit.unit_spec]
Theorem: ZMod.cast_one
File: Mathlib/Data/ZMod/Basic.lean

Proof steps:
state_before: n : ℕ
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
h : m ∣ n
⊢ cast 1 = 1
state_after: case zero
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
h : m ∣ 0
⊢ cast 1 = 1

case succ
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
n : ℕ
h : m ∣ n + 1
⊢ cast 1 = 1
tactic: cases' n with n
state_before: case succ
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
n : ℕ
h : m ∣ n + 1
⊢ cast 1 = 1
state_after: case succ
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
n : ℕ
h : m ∣ n + 1
⊢ ↑(1 % (n + 1)) = 1
tactic: show ((1 % (n + 1) : ℕ) : R) = 1
state_before: case succ
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
n : ℕ
h : m ∣ n + 1
⊢ ↑(1 % (n + 1)) = 1
state_after: case succ.zero
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
h : m ∣ 0 + 1
⊢ ↑(1 % (0 + 1)) = 1

case succ.succ
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
n✝ : ℕ
h : m ∣ n✝ + 1 + 1
⊢ ↑(1 % (n✝ + 1 + 1)) = 1
tactic: cases n
state_before: case succ.succ
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
n✝ : ℕ
h : m ∣ n✝ + 1 + 1
⊢ ↑(1 % (n✝ + 1 + 1)) = 1
state_after: case succ.succ
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
n✝ : ℕ
h : m ∣ n✝ + 1 + 1
⊢ ↑1 = 1

case succ.succ
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
n✝ : ℕ
h : m ∣ n✝ + 1 + 1
⊢ 1 < n✝ + 1 + 1
tactic: rw [Nat.mod_eq_of_lt]
state_before: case succ.succ
R : Type u_1
inst✝¹ : Ring R
m : ℕ
inst✝ : CharP R m
n✝ : ℕ
h : m ∣ n✝ + 1 + 1
⊢ 1 < n✝ + 1 + 1
state_after: no goals
tactic: exact Nat.lt_of_sub_eq_succ rfl
Theorem: Ideal.span_singleton_mul_right_unit
File: Mathlib/RingTheory/Ideal/Basic.lean

Proof steps:
state_before: α : Type u
β : Type v
a✝ b : α
inst✝ : CommSemiring α
I : Ideal α
a : α
h2 : IsUnit a
x : α
⊢ span {x * a} = span {x}
state_after: no goals
tactic: rw [mul_comm, span_singleton_mul_left_unit h2]
Theorem: MeasureTheory.Measure.measurable_map
File: Mathlib/MeasureTheory/Measure/GiryMonad.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
hf : Measurable f
⊢ Measurable fun μ => map f μ
state_after: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ Measurable fun b => (map f b) s
tactic: refine measurable_of_measurable_coe _ fun s hs => ?_
state_before: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ Measurable fun b => (map f b) s
state_after: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ Measurable fun b => b (f ⁻¹' s)
tactic: simp_rw [map_apply hf hs]
state_before: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ Measurable fun b => b (f ⁻¹' s)
state_after: no goals
tactic: exact measurable_coe (hf hs)
Theorem: Ideal.le_toIdeal_homogeneousHull
File: Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean

Proof steps:
state_before: ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
⊢ I ≤ (homogeneousHull 𝒜 I).toIdeal
state_after: ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
r : A
hr : r ∈ I
⊢ r ∈ (homogeneousHull 𝒜 I).toIdeal
tactic: intro r hr
state_before: ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
r : A
hr : r ∈ I
⊢ r ∈ (homogeneousHull 𝒜 I).toIdeal
state_after: no goals
tactic: classical
  rw [← DirectSum.sum_support_decompose 𝒜 r]
  refine Ideal.sum_mem _ ?_
  intro j _
  apply Ideal.subset_span
  use j
  use ⟨r, hr⟩
Theorem: ENNReal.rpow_lt_top_iff_of_pos
File: Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean

Proof steps:
state_before: x : ℝ≥0∞
y : ℝ
hy : 0 < y
⊢ x ^ y < ⊤ ↔ x < ⊤
state_after: no goals
tactic: simp only [lt_top_iff_ne_top, Ne, rpow_eq_top_iff_of_pos hy]
Theorem: RatFunc.mk_eq_mk
File: Mathlib/FieldTheory/RatFunc/Defs.lean

Proof steps:
state_before: K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
p q p' q' : K[X]
hq : q ≠ 0
hq' : q' ≠ 0
⊢ RatFunc.mk p q = RatFunc.mk p' q' ↔ p * q' = p' * q
state_after: no goals
tactic: rw [mk_def_of_ne _ hq, mk_def_of_ne _ hq', ofFractionRing_injective.eq_iff,
    IsLocalization.mk'_eq_iff_eq', -- Porting note: removed `[anonymous], [anonymous]`
    (IsFractionRing.injective K[X] (FractionRing K[X])).eq_iff]
Theorem: round_sub_int
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
x : α
y : ℤ
⊢ round (x - ↑y) = round x - y
state_after: F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
x : α
y : ℤ
⊢ round (x + -↑y) = round x - y
tactic: rw [sub_eq_add_neg]
state_before: F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
x : α
y : ℤ
⊢ round (x + -↑y) = round x - y
state_after: F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
x : α
y : ℤ
⊢ round (x + ↑(-y)) = round x - y
tactic: norm_cast
state_before: F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
x : α
y : ℤ
⊢ round (x + ↑(-y)) = round x - y
state_after: no goals
tactic: rw [round_add_int, sub_eq_add_neg]
Theorem: ContinuousAt.finset_sup'
File: Mathlib/Topology/Order/Lattice.lean

Proof steps:
state_before: L : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace L
inst✝² : TopologicalSpace X
ι : Type u_3
inst✝¹ : SemilatticeSup L
inst✝ : ContinuousSup L
s : Finset ι
f : ι → X → L
t : Set X
x : X
hne : s.Nonempty
hs : ∀ i ∈ s, ContinuousAt (f i) x
⊢ ContinuousAt (s.sup' hne f) x
state_after: no goals
tactic: simpa only [← Finset.sup'_apply] using finset_sup'_apply hne hs
Theorem: Finset.image₂_image_left
File: Mathlib/Data/Finset/NAry.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
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
inst✝⁷ : DecidableEq α'
inst✝⁶ : DecidableEq β'
inst✝⁵ : DecidableEq γ
inst✝⁴ : DecidableEq γ'
inst✝³ : DecidableEq δ
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
inst✝ : DecidableEq ε'
f✝ f' : α → β → γ
g✝ g' : α → β → γ → δ
s s' : Finset α
t t' : Finset β
u u' : Finset γ
a a' : α
b b' : β
c : γ
f : γ → β → δ
g : α → γ
⊢ ↑(image₂ f (image g s) t) = ↑(image₂ (fun a b => f (g a) b) s t)
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
inst✝⁷ : DecidableEq α'
inst✝⁶ : DecidableEq β'
inst✝⁵ : DecidableEq γ
inst✝⁴ : DecidableEq γ'
inst✝³ : DecidableEq δ
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
inst✝ : DecidableEq ε'
f✝ f' : α → β → γ
g✝ g' : α → β → γ → δ
s s' : Finset α
t t' : Finset β
u u' : Finset γ
a a' : α
b b' : β
c : γ
f : γ → β → δ
g : α → γ
⊢ image2 f (g '' ↑s) ↑t = image2 (fun a b => f (g a) b) ↑s ↑t
tactic: push_cast
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
inst✝⁷ : DecidableEq α'
inst✝⁶ : DecidableEq β'
inst✝⁵ : DecidableEq γ
inst✝⁴ : DecidableEq γ'
inst✝³ : DecidableEq δ
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
inst✝ : DecidableEq ε'
f✝ f' : α → β → γ
g✝ g' : α → β → γ → δ
s s' : Finset α
t t' : Finset β
u u' : Finset γ
a a' : α
b b' : β
c : γ
f : γ → β → δ
g : α → γ
⊢ image2 f (g '' ↑s) ↑t = image2 (fun a b => f (g a) b) ↑s ↑t
state_after: no goals
tactic: exact image2_image_left _ _
Theorem: Equiv.subLeft_apply
File: Mathlib/Algebra/Group/Units/Equiv.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
A : Type u_4
B : Type u_5
M : Type u_6
N : Type u_7
P : Type u_8
Q : Type u_9
G : Type u_10
H : Type u_11
inst✝ : Group G
a b : G
⊢ (fun b => b⁻¹ * a) ((fun b => a / b) b) = b
state_after: no goals
tactic: simp [div_eq_mul_inv]
state_before: F : Type u_1
α : Type u_2
β : Type u_3
A : Type u_4
B : Type u_5
M : Type u_6
N : Type u_7
P : Type u_8
Q : Type u_9
G : Type u_10
H : Type u_11
inst✝ : Group G
a b : G
⊢ (fun b => a / b) ((fun b => b⁻¹ * a) b) = b
state_after: no goals
tactic: simp [div_eq_mul_inv]
Theorem: List.IsPrefix.filter
File: .lake/packages/batteries/Batteries/Data/List/Lemmas.lean

Proof steps:
state_before: α : Type u_1
p : α → Bool
l₁ l₂ : List α
h : l₁ <+: l₂
⊢ List.filter p l₁ <+: List.filter p l₂
state_after: case intro
α : Type u_1
p : α → Bool
l₁ xs : List α
⊢ List.filter p l₁ <+: List.filter p (l₁ ++ xs)
tactic: obtain ⟨xs, rfl⟩ := h
state_before: case intro
α : Type u_1
p : α → Bool
l₁ xs : List α
⊢ List.filter p l₁ <+: List.filter p (l₁ ++ xs)
state_after: case intro
α : Type u_1
p : α → Bool
l₁ xs : List α
⊢ List.filter p l₁ <+: List.filter p l₁ ++ List.filter p xs
tactic: rw [filter_append]
state_before: case intro
α : Type u_1
p : α → Bool
l₁ xs : List α
⊢ List.filter p l₁ <+: List.filter p l₁ ++ List.filter p xs
state_after: no goals
tactic: apply prefix_append
Theorem: Linarith.lt_of_eq_of_lt
File: Mathlib/Tactic/Linarith/Lemmas.lean

Proof steps:
state_before: α : Type u_1
inst✝ : OrderedSemiring α
a b : α
ha : a = 0
hb : b < 0
⊢ a + b < 0
state_after: no goals
tactic: simp [*]
Theorem: PseudoMetricSpace.replaceTopology_eq
File: Mathlib/Topology/MetricSpace/Pseudo/Defs.lean

Proof steps:
state_before: α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
γ : Type u_3
U : TopologicalSpace γ
m : PseudoMetricSpace γ
H : U = UniformSpace.toTopologicalSpace
⊢ m.replaceTopology H = m
state_after: case h.dist.h.h
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
γ : Type u_3
U : TopologicalSpace γ
m : PseudoMetricSpace γ
H : U = UniformSpace.toTopologicalSpace
x✝¹ x✝ : γ
⊢ dist x✝¹ x✝ = dist x✝¹ x✝
tactic: ext
state_before: case h.dist.h.h
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
γ : Type u_3
U : TopologicalSpace γ
m : PseudoMetricSpace γ
H : U = UniformSpace.toTopologicalSpace
x✝¹ x✝ : γ
⊢ dist x✝¹ x✝ = dist x✝¹ x✝
state_after: no goals
tactic: rfl
Theorem: ENNReal.lintegral_rpow_funMulInvSnorm_eq_one
File: Mathlib/MeasureTheory/Integral/MeanInequalities.lean

Proof steps:
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0_lt : 0 < p
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
⊢ ∫⁻ (c : α), funMulInvSnorm f p μ c ^ p ∂μ = 1
state_after: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0_lt : 0 < p
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
⊢ ∫⁻ (c : α), f c ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ ∂μ = 1
tactic: simp_rw [funMulInvSnorm_rpow hp0_lt]
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0_lt : 0 < p
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
⊢ ∫⁻ (c : α), f c ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ ∂μ = 1
state_after: case hr
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0_lt : 0 < p
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
⊢ (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ ≠ ⊤
tactic: rw [lintegral_mul_const', ENNReal.mul_inv_cancel hf_nonzero hf_top]
state_before: case hr
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0_lt : 0 < p
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
⊢ (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ ≠ ⊤
state_after: no goals
tactic: rwa [inv_ne_top]
Theorem: Sum.isLeft_iff
File: .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean

Proof steps:
state_before: α✝ : Type u_1
β✝ : Type u_2
x : α✝ ⊕ β✝
⊢ x.isLeft = true ↔ ∃ y, x = inl y
state_after: no goals
tactic: cases x <;> simp
Theorem: IsLocalMin.hasDerivAt_eq_zero
File: Mathlib/Analysis/Calculus/LocalExtr/Basic.lean

Proof steps:
state_before: f : ℝ → ℝ
f' : ℝ
s : Set ℝ
a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ f' = 0
state_after: no goals
tactic: simpa using DFunLike.congr_fun (h.hasFDerivAt_eq_zero (hasDerivAt_iff_hasFDerivAt.1 hf)) 1
Theorem: FirstOrder.Language.Term.realize_restrictVar
File: Mathlib/ModelTheory/Semantics.lean

Proof steps:
state_before: L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝³ : L.Structure M
inst✝² : L.Structure N
inst✝¹ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
inst✝ : DecidableEq α
t : L.Term α
s : Set α
h : ↑t.varFinset ⊆ s
v : α → M
⊢ realize (v ∘ Subtype.val) (t.restrictVar (Set.inclusion h)) = realize v t
state_after: case var
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝³ : L.Structure M
inst✝² : L.Structure N
inst✝¹ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
inst✝ : DecidableEq α
s : Set α
v : α → M
a✝ : α
h : ↑(var a✝).varFinset ⊆ s
⊢ realize (v ∘ Subtype.val) ((var a✝).restrictVar (Set.inclusion h)) = realize v (var a✝)

case func
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝³ : L.Structure M
inst✝² : L.Structure N
inst✝¹ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
inst✝ : DecidableEq α
s : Set α
v : α → M
l✝ : ℕ
_f✝ : L.Functions l✝
_ts✝ : Fin l✝ → L.Term α
ih :
  ∀ (a : Fin l✝) (h : ↑(_ts✝ a).varFinset ⊆ s),
    realize (v ∘ Subtype.val) ((_ts✝ a).restrictVar (Set.inclusion h)) = realize v (_ts✝ a)
h : ↑(func _f✝ _ts✝).varFinset ⊆ s
⊢ realize (v ∘ Subtype.val) ((func _f✝ _ts✝).restrictVar (Set.inclusion h)) = realize v (func _f✝ _ts✝)
tactic: induction' t with _ _ _ _ ih
Theorem: IsUpperSet.isLowerSet_preimage_coe
File: Mathlib/Order/UpperLower/Basic.lean

Proof steps:
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 : LE α
inst✝ : LE β
s t : Set α
a : α
hs : IsUpperSet s
⊢ IsLowerSet (Subtype.val ⁻¹' t) ↔ ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t
state_after: no goals
tactic: aesop
Theorem: mul_eq_of_eq_inv_mul₀
File: Mathlib/Algebra/GroupWithZero/Units/Basic.lean

Proof steps:
state_before: α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a b c d : G₀
m n : ℕ
hb : b ≠ 0
h : b = a⁻¹ * c
⊢ a * b = c
state_after: α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a b c d : G₀
m n : ℕ
hb : b ≠ 0
h : b = a⁻¹ * c
⊢ a ≠ 0
tactic: rwa [← eq_inv_mul_iff_mul_eq₀]
state_before: α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a b c d : G₀
m n : ℕ
hb : b ≠ 0
h : b = a⁻¹ * c
⊢ a ≠ 0
state_after: α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
b c d : G₀
m n : ℕ
hb : b ≠ 0
h : b = 0⁻¹ * c
⊢ False
tactic: rintro rfl
state_before: α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
b c d : G₀
m n : ℕ
hb : b ≠ 0
h : b = 0⁻¹ * c
⊢ False
state_after: no goals
tactic: simp [hb] at h
Theorem: Real.iInf_mul_of_nonneg
File: Mathlib/Data/Real/Pointwise.lean

Proof steps:
state_before: ι : Sort u_1
α : Type u_2
inst✝ : LinearOrderedField α
r : ℝ
ha : 0 ≤ r
f : ι → ℝ
⊢ (⨅ i, f i) * r = ⨅ i, f i * r
state_after: no goals
tactic: simp only [Real.mul_iInf_of_nonneg ha, mul_comm]
Theorem: IsLowerSet.memberSubfamily
File: Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean

Proof steps:
state_before: α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
s : Finset α
a : α
h : IsLowerSet ↑𝒜
⊢ IsLowerSet ↑(Finset.memberSubfamily a 𝒜)
state_after: α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
s✝ : Finset α
a : α
h : IsLowerSet ↑𝒜
s t : Finset α
hts : t ≤ s
⊢ s ∈ ↑(Finset.memberSubfamily a 𝒜) → t ∈ ↑(Finset.memberSubfamily a 𝒜)
tactic: rintro s t hts
state_before: α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
s✝ : Finset α
a : α
h : IsLowerSet ↑𝒜
s t : Finset α
hts : t ≤ s
⊢ s ∈ ↑(Finset.memberSubfamily a 𝒜) → t ∈ ↑(Finset.memberSubfamily a 𝒜)
state_after: α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
s✝ : Finset α
a : α
h : IsLowerSet ↑𝒜
s t : Finset α
hts : t ≤ s
⊢ insert a s ∈ 𝒜 ∧ a ∉ s → insert a t ∈ 𝒜 ∧ a ∉ t
tactic: simp_rw [mem_coe, mem_memberSubfamily]
state_before: α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
s✝ : Finset α
a : α
h : IsLowerSet ↑𝒜
s t : Finset α
hts : t ≤ s
⊢ insert a s ∈ 𝒜 ∧ a ∉ s → insert a t ∈ 𝒜 ∧ a ∉ t
state_after: no goals
tactic: exact And.imp (h <| insert_subset_insert _ hts) (mt <| @hts _)
Theorem: ENNReal.iUnion_Ioc_coe_nat
File: Mathlib/Data/ENNReal/Basic.lean

Proof steps:
state_before: α : Type u_1
a b c d : ℝ≥0∞
r p q : ℝ≥0
⊢ ⋃ n, Ioc a ↑n = Ioi a \ {⊤}
state_after: no goals
tactic: simp only [← Ioi_inter_Iic, ← inter_iUnion, iUnion_Iic_coe_nat, diff_eq]
Theorem: CategoryTheory.Adjunction.eq_homEquiv_apply
File: Mathlib/CategoryTheory/Adjunction/Basic.lean

Proof steps:
state_before: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
X' X : C
Y Y' : D
A : C
B : D
f : F.obj A ⟶ B
g : A ⟶ G.obj B
h : g = (adj.homEquiv A B) f
⊢ (adj.homEquiv A B).symm g = f
state_after: case refl
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
X' X : C
Y Y' : D
A : C
B : D
f : F.obj A ⟶ B
⊢ (adj.homEquiv A B).symm ((adj.homEquiv A B) f) = f
tactic: cases h
state_before: case refl
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
X' X : C
Y Y' : D
A : C
B : D
f : F.obj A ⟶ B
⊢ (adj.homEquiv A B).symm ((adj.homEquiv A B) f) = f
state_after: no goals
tactic: simp
state_before: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
X' X : C
Y Y' : D
A : C
B : D
f : F.obj A ⟶ B
g : A ⟶ G.obj B
h : (adj.homEquiv A B).symm g = f
⊢ g = (adj.homEquiv A B) f
state_after: case refl
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
X' X : C
Y Y' : D
A : C
B : D
g : A ⟶ G.obj B
⊢ g = (adj.homEquiv A B) ((adj.homEquiv A B).symm g)
tactic: cases h
state_before: case refl
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
X' X : C
Y Y' : D
A : C
B : D
g : A ⟶ G.obj B
⊢ g = (adj.homEquiv A B) ((adj.homEquiv A B).symm g)
state_after: no goals
tactic: simp
Theorem: IsAlgClosed.exists_eval₂_eq_zero_of_injective
File: Mathlib/FieldTheory/IsAlgClosed/Basic.lean

Proof steps:
state_before: k : Type u
inst✝² : Field k
R : Type u_1
inst✝¹ : Ring R
inst✝ : IsAlgClosed k
f : R →+* k
hf : Function.Injective ⇑f
p : R[X]
hp : p.degree ≠ 0
⊢ (map f p).degree ≠ 0
state_after: no goals
tactic: rwa [degree_map_eq_of_injective hf]
state_before: k : Type u
inst✝² : Field k
R : Type u_1
inst✝¹ : Ring R
inst✝ : IsAlgClosed k
f : R →+* k
hf : Function.Injective ⇑f
p : R[X]
hp : p.degree ≠ 0
x : k
hx : (map f p).IsRoot x
⊢ eval₂ f x p = 0
state_after: no goals
tactic: rwa [eval₂_eq_eval_map, ← IsRoot]
Theorem: StructureGroupoid.LocalInvariantProp.liftPropAt_of_mem_maximalAtlas
File: Mathlib/Geometry/Manifold/LocalInvariantProperties.lean

Proof steps:
state_before: H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
X : Type u_5
inst✝⁷ : TopologicalSpace H
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : TopologicalSpace H'
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
inst✝¹ : TopologicalSpace X
G : StructureGroupoid H
G' : StructureGroupoid H'
e e' : PartialHomeomorph M H
f f' : PartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g g' : M → M'
s t : Set M
x : M
Q : (H → H) → Set H → H → Prop
hG✝ : G.LocalInvariantProp G' P
inst✝ : HasGroupoid M G
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
hx : x ∈ e.source
⊢ LiftPropAt Q (↑e) x
state_after: H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
X : Type u_5
inst✝⁷ : TopologicalSpace H
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : TopologicalSpace H'
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
inst✝¹ : TopologicalSpace X
G : StructureGroupoid H
G' : StructureGroupoid H'
e e' : PartialHomeomorph M H
f f' : PartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g g' : M → M'
s t : Set M
x : M
Q : (H → H) → Set H → H → Prop
hG✝ : G.LocalInvariantProp G' P
inst✝ : HasGroupoid M G
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
hx : x ∈ e.source
⊢ Q (↑(PartialHomeomorph.refl H) ∘ ↑e ∘ ↑e.symm) (↑e.symm ⁻¹' univ) (↑e x)
tactic: simp_rw [LiftPropAt, hG.liftPropWithinAt_indep_chart he hx G.id_mem_maximalAtlas (mem_univ _),
    (e.continuousAt hx).continuousWithinAt, true_and_iff]
state_before: H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
X : Type u_5
inst✝⁷ : TopologicalSpace H
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : TopologicalSpace H'
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
inst✝¹ : TopologicalSpace X
G : StructureGroupoid H
G' : StructureGroupoid H'
e e' : PartialHomeomorph M H
f f' : PartialHomeomorph M' H'
P : (H → H') → Set H → H → Prop
g g' : M → M'
s t : Set M
x : M
Q : (H → H) → Set H → H → Prop
hG✝ : G.LocalInvariantProp G' P
inst✝ : HasGroupoid M G
hG : G.LocalInvariantProp G Q
hQ : ∀ (y : H), Q id univ y
he : e ∈ maximalAtlas M G
hx : x ∈ e.source
⊢ Q (↑(PartialHomeomorph.refl H) ∘ ↑e ∘ ↑e.symm) (↑e.symm ⁻¹' univ) (↑e x)
state_after: no goals
tactic: exact hG.congr' (e.eventually_right_inverse' hx) (hQ _)
Theorem: Rat.neg_def
File: Mathlib/Data/Rat/Defs.lean

Proof steps:
state_before: q✝ q : ℚ
⊢ -q = -q.num /. ↑q.den
state_after: no goals
tactic: rw [← neg_divInt, num_divInt_den]