1337xyz1337xyz/leandojo-benchmark4-v10 huggingface dataset:
Theorem: RingCon.sSup_def
File: Mathlib/RingTheory/Congruence/Basic.lean

tactic: rw [sSup_eq_ringConGen, sSup_image]
state_before: α : Type u_1
R : Type u_2
inst✝¹ : Add R
inst✝ : Mul R
S : Set (RingCon R)
⊢ sSup S = ringConGen (sSup (DFunLike.coe '' S))
state_after: α : Type u_1
R : Type u_2
inst✝¹ : Add R
inst✝ : Mul R
S : Set (RingCon R)
⊢ (ringConGen fun x y => ∃ c ∈ S, c x y) = ringConGen (⨆ a ∈ S, ⇑a)

tactic: congr with (x y)
state_before: α : Type u_1
R : Type u_2
inst✝¹ : Add R
inst✝ : Mul R
S : Set (RingCon R)
⊢ (ringConGen fun x y => ∃ c ∈ S, c x y) = ringConGen (⨆ a ∈ S, ⇑a)
state_after: case e_r.h.h.a
α : Type u_1
R : Type u_2
inst✝¹ : Add R
inst✝ : Mul R
S : Set (RingCon R)
x y : R
⊢ (∃ c ∈ S, c x y) ↔ (⨆ a ∈ S, ⇑a) x y

tactic: simp only [sSup_image, iSup_apply, iSup_Prop_eq, exists_prop, rel_eq_coe]
state_before: case e_r.h.h.a
α : Type u_1
R : Type u_2
inst✝¹ : Add R
inst✝ : Mul R
S : Set (RingCon R)
x y : R
⊢ (∃ c ∈ S, c x y) ↔ (⨆ a ∈ S, ⇑a) x y
state_after: no goals
Theorem: MeasureTheory.Measure.pi_pi
File: Mathlib/MeasureTheory/Constructions/Pi.lean

tactic: haveI : Encodable ι := Fintype.toEncodable ι
state_before: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
⊢ (Measure.pi μ) (univ.pi s) = ∏ i : ι, (μ i) (s i)
state_after: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
this : Encodable ι
⊢ (Measure.pi μ) (univ.pi s) = ∏ i : ι, (μ i) (s i)

tactic: rw [← pi'_eq_pi, pi'_pi]
state_before: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
this : Encodable ι
⊢ (Measure.pi μ) (univ.pi s) = ∏ i : ι, (μ i) (s i)
state_after: no goals
Theorem: Finset.max'_image
File: Mathlib/Data/Finset/Lattice.lean

tactic: simp only [max', sup'_image]
state_before: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
ι : Type u_5
κ : Type u_6
inst✝¹ : LinearOrder α
s✝ : Finset α
H : s✝.Nonempty
x : α
inst✝ : LinearOrder β
f : α → β
hf : Monotone f
s : Finset α
h : (image f s).Nonempty
⊢ (image f s).max' h = f (s.max' ⋯)
state_after: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
ι : Type u_5
κ : Type u_6
inst✝¹ : LinearOrder α
s✝ : Finset α
H : s✝.Nonempty
x : α
inst✝ : LinearOrder β
f : α → β
hf : Monotone f
s : Finset α
h : (image f s).Nonempty
⊢ s.sup' ⋯ (id ∘ f) = f (s.sup' ⋯ id)

tactic: exact .symm <| comp_sup'_eq_sup'_comp _ _ fun _ _ ↦ hf.map_max
state_before: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
ι : Type u_5
κ : Type u_6
inst✝¹ : LinearOrder α
s✝ : Finset α
H : s✝.Nonempty
x : α
inst✝ : LinearOrder β
f : α → β
hf : Monotone f
s : Finset α
h : (image f s).Nonempty
⊢ s.sup' ⋯ (id ∘ f) = f (s.sup' ⋯ id)
state_after: no goals
Theorem: Matrix.cons_add_cons
File: Mathlib/Data/Fin/VecNotation.lean

tactic: simp
state_before: α : Type u
m n o : ℕ
m' : Type u_1
n' : Type u_2
o' : Type u_3
inst✝ : Add α
x : α
v : Fin n → α
y : α
w : Fin n → α
⊢ vecCons x v + vecCons y w = vecCons (x + y) (v + w)
state_after: no goals
Theorem: MeasurableEquiv.coe_sumPiEquivProdPi_symm
File: Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean

tactic: rfl
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α✝
inst✝⁶ : MeasurableSpace α✝
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
inst✝³ : MeasurableSpace δ
π : δ' → Type u_6
π' : δ' → Type u_7
inst✝² : (x : δ') → MeasurableSpace (π x)
inst✝¹ : (x : δ') → MeasurableSpace (π' x)
α : δ ⊕ δ' → Type u_8
inst✝ : (i : δ ⊕ δ') → MeasurableSpace (α i)
⊢ ⇑(sumPiEquivProdPi α).symm = ⇑(Equiv.sumPiEquivProdPi α).symm
state_after: no goals
Theorem: IsOfFinOrder.finite_powers
File: Mathlib/GroupTheory/OrderOfElement.lean

tactic: classical rw [ha.powers_eq_image_range_orderOf]; exact Finset.finite_toSet _
state_before: G : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
inst✝ : Monoid G
a b x y : G
n m : ℕ
ha : IsOfFinOrder a
⊢ (↑(powers a)).Finite
state_after: no goals

tactic: rw [ha.powers_eq_image_range_orderOf]
state_before: G : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
inst✝ : Monoid G
a b x y : G
n m : ℕ
ha : IsOfFinOrder a
⊢ (↑(powers a)).Finite
state_after: G : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
inst✝ : Monoid G
a b x y : G
n m : ℕ
ha : IsOfFinOrder a
⊢ (↑(Finset.image (fun x => a ^ x) (Finset.range (orderOf a)))).Finite

tactic: exact Finset.finite_toSet _
state_before: G : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
inst✝ : Monoid G
a b x y : G
n m : ℕ
ha : IsOfFinOrder a
⊢ (↑(Finset.image (fun x => a ^ x) (Finset.range (orderOf a)))).Finite
state_after: no goals
Theorem: List.Duplicate.of_duplicate_cons
File: Mathlib/Data/List/Duplicate.lean

tactic: simpa [duplicate_cons_iff, hx.symm] using h
state_before: α : Type u_1
l : List α
x y : α
h : x ∈+ y :: l
hx : x ≠ y
⊢ x ∈+ l
state_after: no goals
Theorem: TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne
File: Mathlib/Topology/Separation.lean

tactic: rcases hb.isOpen_iff.1 isOpen_ne x h with ⟨a, ab, xa, ha⟩
state_before: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
b : Set (Set X)
hb : IsTopologicalBasis b
x y : X
h : x ≠ y
⊢ ∃ a ∈ b, x ∈ a ∧ y ∉ a
state_after: case intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
b : Set (Set X)
hb : IsTopologicalBasis b
x y : X
h : x ≠ y
a : Set X
ab : a ∈ b
xa : x ∈ a
ha : a ⊆ {y_1 | y_1 ≠ y}
⊢ ∃ a ∈ b, x ∈ a ∧ y ∉ a

tactic: exact ⟨a, ab, xa, fun h => ha h rfl⟩
state_before: case intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
b : Set (Set X)
hb : IsTopologicalBasis b
x y : X
h : x ≠ y
a : Set X
ab : a ∈ b
xa : x ∈ a
ha : a ⊆ {y_1 | y_1 ≠ y}
⊢ ∃ a ∈ b, x ∈ a ∧ y ∉ a
state_after: no goals
Theorem: Associates.prime_pow_le_iff_le_bcount
File: Mathlib/RingTheory/UniqueFactorizationDomain.lean

tactic: rcases Associates.exists_non_zero_rep h₁ with ⟨m, hm, rfl⟩
state_before: α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : DecidableEq (Associates α)
m p : Associates α
h₁ : m ≠ 0
h₂ : Irreducible p
k : ℕ
⊢ p ^ k ≤ m ↔ k ≤ bcount ⟨p, h₂⟩ m.factors
state_after: case intro.intro
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : DecidableEq (Associates α)
p : Associates α
h₂ : Irreducible p
k : ℕ
m : α
hm : m ≠ 0
h₁ : Associates.mk m ≠ 0
⊢ p ^ k ≤ Associates.mk m ↔ k ≤ bcount ⟨p, h₂⟩ (Associates.mk m).factors

tactic: have := nontrivial_of_ne _ _ hm
state_before: case intro.intro
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : DecidableEq (Associates α)
p : Associates α
h₂ : Irreducible p
k : ℕ
m : α
hm : m ≠ 0
h₁ : Associates.mk m ≠ 0
⊢ p ^ k ≤ Associates.mk m ↔ k ≤ bcount ⟨p, h₂⟩ (Associates.mk m).factors
state_after: case intro.intro
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : DecidableEq (Associates α)
p : Associates α
h₂ : Irreducible p
k : ℕ
m : α
hm : m ≠ 0
h₁ : Associates.mk m ≠ 0
this : Nontrivial α
⊢ p ^ k ≤ Associates.mk m ↔ k ≤ bcount ⟨p, h₂⟩ (Associates.mk m).factors

tactic: rw [bcount, factors_mk, Multiset.le_count_iff_replicate_le, ← factors_le, factors_prime_pow,
  factors_mk, WithTop.coe_le_coe] <;> assumption
state_before: case intro.intro
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : DecidableEq (Associates α)
p : Associates α
h₂ : Irreducible p
k : ℕ
m : α
hm : m ≠ 0
h₁ : Associates.mk m ≠ 0
this : Nontrivial α
⊢ p ^ k ≤ Associates.mk m ↔ k ≤ bcount ⟨p, h₂⟩ (Associates.mk m).factors
state_after: no goals
Theorem: dense_discrete
File: Mathlib/Topology/Order.lean

tactic: simp [dense_iff_closure_eq]
state_before: α : Type u_1
t t₁ t₂ : TopologicalSpace α
s✝ : Set α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
s : Set α
⊢ Dense s ↔ s = univ
state_after: no goals
Theorem: Nat.gcd_sub_self_right
File: Mathlib/Data/Nat/GCD/Basic.lean

tactic: rw [gcd_comm, gcd_sub_self_left h, gcd_comm]
state_before: m n : ℕ
h : m ≤ n
⊢ m.gcd (n - m) = m.gcd n
state_after: no goals
Theorem: intervalIntegral.norm_integral_le_integral_norm
File: Mathlib/MeasureTheory/Integral/IntervalIntegral.lean

tactic: rw [uIoc_of_le h, integral_of_le h]
state_before: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a b : ℝ
f g : ℝ → E
μ : Measure ℝ
h : a ≤ b
⊢ ∫ (x : ℝ) in Ι a b, ‖f x‖ ∂μ = ∫ (x : ℝ) in a..b, ‖f x‖ ∂μ
state_after: no goals
Theorem: MonoidHom.ker_prodMap
File: Mathlib/Algebra/Group/Subgroup/Basic.lean

tactic: rw [← comap_bot, ← comap_bot, ← comap_bot, ← prodMap_comap_prod, bot_prod_bot]
state_before: G : Type u_1
G'✝ : Type u_2
G'' : Type u_3
inst✝⁸ : Group G
inst✝⁷ : Group G'✝
inst✝⁶ : Group G''
A : Type u_4
inst✝⁵ : AddGroup A
N : Type u_5
P : Type u_6
inst✝⁴ : Group N
inst✝³ : Group P
K : Subgroup G
M : Type u_7
inst✝² : MulOneClass M
G' : Type u_8
N' : Type u_9
inst✝¹ : Group G'
inst✝ : Group N'
f : G →* N
g : G' →* N'
⊢ (f.prodMap g).ker = f.ker.prod g.ker
state_after: no goals
Theorem: String.data_takeWhile
File: .lake/packages/batteries/Batteries/Data/String/Lemmas.lean

tactic: rw [takeWhile_eq]
state_before: p : Char → Bool
s : String
⊢ (s.takeWhile p).data = List.takeWhile p s.data
state_after: no goals
Theorem: affineIndependent_iff_not_collinear_set
File: Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean

tactic: rw [affineIndependent_iff_not_collinear]
state_before: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
⊢ AffineIndependent k ![p₁, p₂, p₃] ↔ ¬Collinear k {p₁, p₂, p₃}
state_after: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
⊢ ¬Collinear k (Set.range ![p₁, p₂, p₃]) ↔ ¬Collinear k {p₁, p₂, p₃}

tactic: simp_rw [Matrix.range_cons, Matrix.range_empty, Set.singleton_union, insert_emptyc_eq]
state_before: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
⊢ ¬Collinear k (Set.range ![p₁, p₂, p₃]) ↔ ¬Collinear k {p₁, p₂, p₃}
state_after: no goals
Theorem: MeasureTheory.AEDisjoint.iUnion_left_iff
File: Mathlib/MeasureTheory/Measure/AEDisjoint.lean

tactic: simp only [AEDisjoint, iUnion_inter, measure_iUnion_null_iff]
state_before: ι : Type u_1
α : Type u_2
m : MeasurableSpace α
μ : Measure α
s✝ t u v : Set α
inst✝ : Countable ι
s : ι → Set α
⊢ AEDisjoint μ (⋃ i, s i) t ↔ ∀ (i : ι), AEDisjoint μ (s i) t
state_after: no goals
Theorem: Finset.smul_zero_subset
File: Mathlib/Data/Finset/Pointwise.lean

tactic: simp [subset_iff, mem_smul]
state_before: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : Zero β
inst✝¹ : SMulZeroClass α β
inst✝ : DecidableEq β
s✝ : Finset α
t : Finset β
a : α
s : Finset α
⊢ s • 0 ⊆ 0
state_after: no goals
Theorem: Commute.mul_neg_geom_sum₂
File: Mathlib/Algebra/GeomSum.lean

tactic: apply op_injective
state_before: α : Type u
inst✝ : Ring α
x y : α
h : Commute x y
n : ℕ
⊢ (y - x) * ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i) = y ^ n - x ^ n
state_after: case a
α : Type u
inst✝ : Ring α
x y : α
h : Commute x y
n : ℕ
⊢ MulOpposite.op ((y - x) * ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) = MulOpposite.op (y ^ n - x ^ n)

tactic: simp only [op_mul, op_sub, op_geom_sum₂, op_pow]
state_before: case a
α : Type u
inst✝ : Ring α
x y : α
h : Commute x y
n : ℕ
⊢ MulOpposite.op ((y - x) * ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) = MulOpposite.op (y ^ n - x ^ n)
state_after: case a
α : Type u
inst✝ : Ring α
x y : α
h : Commute x y
n : ℕ
⊢ MulOpposite.op (∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) * (MulOpposite.op y - MulOpposite.op x) =
    MulOpposite.op y ^ n - MulOpposite.op x ^ n

tactic: simp [(Commute.op h.symm).geom_sum₂_mul n]
state_before: case a
α : Type u
inst✝ : Ring α
x y : α
h : Commute x y
n : ℕ
⊢ MulOpposite.op (∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) * (MulOpposite.op y - MulOpposite.op x) =
    MulOpposite.op y ^ n - MulOpposite.op x ^ n
state_after: no goals
Theorem: FractionalIdeal.zero_le
File: Mathlib/RingTheory/FractionalIdeal/Basic.lean

tactic: intro x hx
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 : FractionalIdeal S P
⊢ 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 : FractionalIdeal S P
x : P
hx : x ∈ 0
⊢ x ∈ I

tactic: rw [(mem_zero_iff _).mp hx]
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 : FractionalIdeal S P
x : P
hx : x ∈ 0
⊢ x ∈ 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 : FractionalIdeal S P
x : P
hx : x ∈ 0
⊢ 0 ∈ I

tactic: exact zero_mem I
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 : FractionalIdeal S P
x : P
hx : x ∈ 0
⊢ 0 ∈ I
state_after: no goals
Theorem: ProbabilityTheory.setIntegral_preCDF_fst
File: Mathlib/Probability/Kernel/Disintegration/CondCdf.lean

tactic: rw [integral_toReal]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ ∫ (x : α) in s, (preCDF ρ r x).toReal ∂ρ.fst = ((ρ.IicSnd ↑r) s).toReal
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ (∫⁻ (a : α) in s, preCDF ρ r a ∂ρ.fst).toReal = ((ρ.IicSnd ↑r) s).toReal

case hfm
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ AEMeasurable (preCDF ρ r) (ρ.fst.restrict s)

case hf
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ ∀ᵐ (x : α) ∂ρ.fst.restrict s, preCDF ρ r x < ⊤

tactic: rw [setLIntegral_preCDF_fst _ _ hs]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ (∫⁻ (a : α) in s, preCDF ρ r a ∂ρ.fst).toReal = ((ρ.IicSnd ↑r) s).toReal
state_after: no goals

tactic: exact measurable_preCDF.aemeasurable
state_before: case hfm
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ AEMeasurable (preCDF ρ r) (ρ.fst.restrict s)
state_after: no goals

tactic: refine ae_restrict_of_ae ?_
state_before: case hf
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ ∀ᵐ (x : α) ∂ρ.fst.restrict s, preCDF ρ r x < ⊤
state_after: case hf
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ ∀ᵐ (x : α) ∂ρ.fst, preCDF ρ r x < ⊤

tactic: filter_upwards [preCDF_le_one ρ] with a ha
state_before: case hf
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ ∀ᵐ (x : α) ∂ρ.fst, preCDF ρ r x < ⊤
state_after: case h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
a : α
ha : ∀ (r : ℚ), preCDF ρ r a ≤ 1
⊢ preCDF ρ r a < ⊤

tactic: exact (ha r).trans_lt ENNReal.one_lt_top
state_before: case h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
a : α
ha : ∀ (r : ℚ), preCDF ρ r a ≤ 1
⊢ preCDF ρ r a < ⊤
state_after: no goals
Theorem: Nat.totient_lt
File: Mathlib/Data/Nat/Totient.lean

tactic: simp [hn.ne', pos_of_gt hn]
state_before: n : ℕ
hn : 1 < n
⊢ 0 ∈ range n ∧ ¬n.Coprime 0
state_after: no goals
Theorem: Affine.Triangle.dist_orthocenter_reflection_circumcenter_finset
File: Mathlib/Geometry/Euclidean/MongePoint.lean

tactic: simp only [mem_singleton, coe_insert, coe_singleton, Set.mem_singleton_iff]
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
i₁ i₂ : Fin 3
h : i₁ ≠ i₂
⊢ dist t.orthocenter ((EuclideanGeometry.reflection (affineSpan ℝ (t.points '' ↑{i₁, i₂}))) (circumcenter t)) =
    circumradius t
state_after: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
i₁ i₂ : Fin 3
h : i₁ ≠ i₂
⊢ dist t.orthocenter ((EuclideanGeometry.reflection (affineSpan ℝ (t.points '' {i₁, i₂}))) (circumcenter t)) =
    circumradius t

tactic: exact dist_orthocenter_reflection_circumcenter _ h
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
i₁ i₂ : Fin 3
h : i₁ ≠ i₂
⊢ dist t.orthocenter ((EuclideanGeometry.reflection (affineSpan ℝ (t.points '' {i₁, i₂}))) (circumcenter t)) =
    circumradius t
state_after: no goals
Theorem: LinearEquiv.lTensor_zpow
File: Mathlib/LinearAlgebra/TensorProduct/Basic.lean

tactic: simpa only [one_zpow] using TensorProduct.congr_zpow (1 : M ≃ₗ[R] M) f n
state_before: R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
g : P ≃ₗ[R] Q
f✝ : N ≃ₗ[R] P
m : M
n✝ : N
p : P
x : M ⊗[R] N
y : N ⊗[R] M
f : N ≃ₗ[R] N
n : ℤ
⊢ lTensor M f ^ n = lTensor M (f ^ n)
state_after: no goals
Theorem: MeasureTheory.Measure.haar.prehaar_nonneg
File: Mathlib/MeasureTheory/Measure/Haar/Basic.lean

tactic: apply div_nonneg <;> norm_cast <;> apply zero_le
state_before: G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ prehaar (↑K₀) U K
state_after: no goals
Theorem: Finset.image₂_image_left
File: Mathlib/Data/Finset/NAry.lean

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 (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: exact image2_image_left _ _
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
Theorem: SubMulAction.orbitRel_of_subMul
File: Mathlib/GroupTheory/GroupAction/SubMulAction.lean

tactic: refine Setoid.ext_iff.2 (fun x y ↦ ?_)
state_before: S : Type u'
T : Type u''
R : Type u
M : Type v
inst✝¹ : Group R
inst✝ : MulAction R M
p : SubMulAction R M
⊢ MulAction.orbitRel R ↥p = Setoid.comap Subtype.val (MulAction.orbitRel R M)
state_after: S : Type u'
T : Type u''
R : Type u
M : Type v
inst✝¹ : Group R
inst✝ : MulAction R M
p : SubMulAction R M
x y : ↥p
⊢ (MulAction.orbitRel R ↥p).Rel x y ↔ (Setoid.comap Subtype.val (MulAction.orbitRel R M)).Rel x y

tactic: rw [Setoid.comap_rel]
state_before: S : Type u'
T : Type u''
R : Type u
M : Type v
inst✝¹ : Group R
inst✝ : MulAction R M
p : SubMulAction R M
x y : ↥p
⊢ (MulAction.orbitRel R ↥p).Rel x y ↔ (Setoid.comap Subtype.val (MulAction.orbitRel R M)).Rel x y
state_after: S : Type u'
T : Type u''
R : Type u
M : Type v
inst✝¹ : Group R
inst✝ : MulAction R M
p : SubMulAction R M
x y : ↥p
⊢ (MulAction.orbitRel R ↥p).Rel x y ↔ (MulAction.orbitRel R M).Rel ↑x ↑y

tactic: exact mem_orbit_subMul_iff
state_before: S : Type u'
T : Type u''
R : Type u
M : Type v
inst✝¹ : Group R
inst✝ : MulAction R M
p : SubMulAction R M
x y : ↥p
⊢ (MulAction.orbitRel R ↥p).Rel x y ↔ (MulAction.orbitRel R M).Rel ↑x ↑y
state_after: no goals
Theorem: ZFSet.toSet_sUnion
File: Mathlib/SetTheory/ZFC/Basic.lean

tactic: ext
state_before: x : ZFSet
⊢ (⋃₀ x).toSet = ⋃₀ (toSet '' x.toSet)
state_after: case h
x x✝ : ZFSet
⊢ x✝ ∈ (⋃₀ x).toSet ↔ x✝ ∈ ⋃₀ (toSet '' x.toSet)

tactic: simp
state_before: case h
x x✝ : ZFSet
⊢ x✝ ∈ (⋃₀ x).toSet ↔ x✝ ∈ ⋃₀ (toSet '' x.toSet)
state_after: no goals