tasksource/leandojo huggingface dataset:
Theorem: isPrimePow_of_minFac_pow_factorization_eq
File: Mathlib/Data/Nat/Factorization/PrimePow.lean

tactic: rcases eq_or_ne n 0 with (rfl | hn')
state_before: R : Type ?u.706
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
h : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n
hn : n ≠ 1
⊢ IsPrimePow n

tactic: refine' ⟨_, _, (Nat.minFac_prime hn).prime, _, h⟩
state_before: case inr
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
h : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n
hn : n ≠ 1
hn' : n ≠ 0
⊢ IsPrimePow n

tactic: rw [pos_iff_ne_zero, ← Finsupp.mem_support_iff, Nat.factor_iff_mem_factorization,
  Nat.mem_factors_iff_dvd hn' (Nat.minFac_prime hn)]
state_before: case inr
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
h : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n
hn : n ≠ 1
hn' : n ≠ 0
⊢ 0 < ↑(Nat.factorization n) (Nat.minFac n)

tactic: apply Nat.minFac_dvd
state_before: case inr
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
h : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n
hn : n ≠ 1
hn' : n ≠ 0
⊢ Nat.minFac n ∣ n

tactic: simp_all
state_before: case inl
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n p : R
k : ℕ
h : Nat.minFac 0 ^ ↑(Nat.factorization 0) (Nat.minFac 0) = 0
hn : 0 ≠ 1
⊢ IsPrimePow 0
Theorem: ContinuousLinearMap.smulRight_one_pow
File: Mathlib/Topology/Algebra/Module/Basic.lean

tactic: induction' n with n ihn
state_before: R : Type u_1
inst✝¹⁵ : Ring R
R₂ : Type ?u.888280
inst✝¹⁴ : Ring R₂
R₃ : Type ?u.888286
inst✝¹³ : Ring R₃
M : Type ?u.888292
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommGroup M
M₂ : Type ?u.888301
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommGroup M₂
M₃ : Type ?u.888310
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommGroup M₃
M₄ : Type ?u.888319
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommGroup M₄
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalRing R
c : R
n : ℕ
⊢ smulRight 1 c ^ n = smulRight 1 (c ^ n)

tactic: ext
state_before: case zero
R : Type u_1
inst✝¹⁵ : Ring R
R₂ : Type ?u.888280
inst✝¹⁴ : Ring R₂
R₃ : Type ?u.888286
inst✝¹³ : Ring R₃
M : Type ?u.888292
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommGroup M
M₂ : Type ?u.888301
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommGroup M₂
M₃ : Type ?u.888310
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommGroup M₃
M₄ : Type ?u.888319
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommGroup M₄
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalRing R
c : R
⊢ smulRight 1 c ^ Nat.zero = smulRight 1 (c ^ Nat.zero)

tactic: simp
state_before: case zero.h
R : Type u_1
inst✝¹⁵ : Ring R
R₂ : Type ?u.888280
inst✝¹⁴ : Ring R₂
R₃ : Type ?u.888286
inst✝¹³ : Ring R₃
M : Type ?u.888292
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommGroup M
M₂ : Type ?u.888301
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommGroup M₂
M₃ : Type ?u.888310
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommGroup M₃
M₄ : Type ?u.888319
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommGroup M₄
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalRing R
c : R
⊢ ↑(smulRight 1 c ^ Nat.zero) 1 = ↑(smulRight 1 (c ^ Nat.zero)) 1

tactic: rw [pow_succ, ihn, mul_def, smulRight_comp, smul_eq_mul, pow_succ']
state_before: case succ
R : Type u_1
inst✝¹⁵ : Ring R
R₂ : Type ?u.888280
inst✝¹⁴ : Ring R₂
R₃ : Type ?u.888286
inst✝¹³ : Ring R₃
M : Type ?u.888292
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommGroup M
M₂ : Type ?u.888301
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommGroup M₂
M₃ : Type ?u.888310
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommGroup M₃
M₄ : Type ?u.888319
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommGroup M₄
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalRing R
c : R
n : ℕ
ihn : smulRight 1 c ^ n = smulRight 1 (c ^ n)
⊢ smulRight 1 c ^ Nat.succ n = smulRight 1 (c ^ Nat.succ n)
Theorem: ProbabilityTheory.cond_univ
File: Mathlib/Probability/ConditionalProbability.lean

tactic: simp [cond, measure_univ, Measure.restrict_univ]
state_before: Ω : Type u_1
m : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
s t : Set Ω
inst✝ : IsProbabilityMeasure μ
⊢ μ[|Set.univ] = μ
Theorem: padicValNat.div'
File: Mathlib/NumberTheory/Padics/PadicVal.lean

tactic: rw [padicValNat.div_of_dvd dvd, eq_zero_of_not_dvd (hp.out.coprime_iff_not_dvd.mp cpm),
  Nat.sub_zero]
state_before: p a b✝ : ℕ
hp : Fact (Nat.Prime p)
m : ℕ
cpm : coprime p m
b : ℕ
dvd : m ∣ b
⊢ padicValNat p (b / m) = padicValNat p b
Theorem: Set.Ico_diff_Iio
File: Mathlib/Data/Set/Intervals/Basic.lean

tactic: rw [diff_eq, compl_Iio, Ico_inter_Ici, sup_eq_max]
state_before: α : Type u_1
β : Type ?u.197269
inst✝¹ : LinearOrder α
inst✝ : LinearOrder β
f : α → β
a a₁ a₂ b b₁ b₂ c d : α
⊢ Ico a b \ Iio c = Ico (max a c) b
Theorem: MeasureTheory.lintegral_zero
File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

tactic: simp
state_before: α : Type u_1
β : Type ?u.74361
γ : Type ?u.74364
δ : Type ?u.74367
m : MeasurableSpace α
μ ν : Measure α
⊢ (∫⁻ (x : α), 0 ∂μ) = 0
Theorem: Matrix.adjugate_pow
File: Mathlib/LinearAlgebra/Matrix/Adjugate.lean

tactic: induction' k with k IH
state_before: m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
k : ℕ
⊢ adjugate (A ^ k) = adjugate A ^ k

tactic: simp
state_before: case zero
m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
⊢ adjugate (A ^ Nat.zero) = adjugate A ^ Nat.zero

tactic: rw [pow_succ', mul_eq_mul, adjugate_mul_distrib, IH, ← mul_eq_mul, pow_succ]
state_before: case succ
m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
k : ℕ
IH : adjugate (A ^ k) = adjugate A ^ k
⊢ adjugate (A ^ Nat.succ k) = adjugate A ^ Nat.succ k
Theorem: LinearMap.flip_separatingLeft
File: Mathlib/LinearAlgebra/SesquilinearForm.lean

tactic: rw [← flip_separatingRight, flip_flip]
state_before: R : Type u_2
R₁ : Type u_1
R₂ : Type u_5
R₃ : Type ?u.540765
M : Type ?u.540768
M₁ : Type u_3
M₂ : Type u_4
Mₗ₁ : Type ?u.540777
Mₗ₁' : Type ?u.540780
Mₗ₂ : Type ?u.540783
Mₗ₂' : Type ?u.540786
K : Type ?u.540789
K₁ : Type ?u.540792
K₂ : Type ?u.540795
V : Type ?u.540798
V₁ : Type ?u.540801
V₂ : Type ?u.540804
n : Type ?u.540807
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring R₁
inst✝⁴ : AddCommMonoid M₁
inst✝³ : Module R₁ M₁
inst✝² : CommSemiring R₂
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R₂ M₂
I₁ : R₁ →+* R
I₂ : R₂ →+* R
I₁' : R₁ →+* R
B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R
⊢ SeparatingLeft (flip B) ↔ SeparatingRight B
Theorem: MeasureTheory.Measure.haar.haarContent_outerMeasure_self_pos
File: Mathlib/MeasureTheory/Measure/Haar/Basic.lean

tactic: refine' zero_lt_one.trans_le _
state_before: G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ 0 < ↑(Content.outerMeasure (haarContent K₀)) ↑K₀

tactic: rw [Content.outerMeasure_eq_iInf]
state_before: G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ 1 ≤ ↑(Content.outerMeasure (haarContent K₀)) ↑K₀

tactic: refine' le_iInf₂ fun U hU => le_iInf fun hK₀ => le_trans _ <| le_iSup₂ K₀.toCompacts hK₀
state_before: G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ 1 ≤
    ⨅ (U : Set G) (hU : IsOpen U) (_ : ↑K₀ ⊆ U), Content.innerContent (haarContent K₀) { carrier := U, is_open' := hU }

tactic: exact haarContent_self.ge
state_before: G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
U : Set G
hU : IsOpen U
hK₀ : ↑K₀ ⊆ U
⊢ 1 ≤ (fun s => ↑(Content.toFun (haarContent K₀) s)) K₀.toCompacts
Theorem: Cardinal.aleph'_omega
File: Mathlib/SetTheory/Cardinal/Ordinal.lean

tactic: simp only [aleph'_le_of_limit omega_isLimit, lt_omega, exists_imp, aleph0_le]
state_before: c : Cardinal
⊢ aleph' ω ≤ c ↔ ℵ₀ ≤ c

tactic: exact forall_swap.trans (forall_congr' fun n => by simp only [forall_eq, aleph'_nat])
state_before: c : Cardinal
⊢ (∀ (o' : Ordinal) (x : ℕ), o' = ↑x → aleph' o' ≤ c) ↔ ∀ (n : ℕ), ↑n ≤ c

tactic: simp only [forall_eq, aleph'_nat]
state_before: c : Cardinal
n : ℕ
⊢ (∀ (x : Ordinal), x = ↑n → aleph' x ≤ c) ↔ ↑n ≤ c
Theorem: CharP.addOrderOf_one
File: Mathlib/Algebra/CharP/Basic.lean

tactic: rw [← Nat.smul_one_eq_coe, addOrderOf_dvd_iff_nsmul_eq_zero]
state_before: R✝ : Type ?u.43695
R : Type u_1
inst✝ : Semiring R
n : ℕ
⊢ ↑n = 0 ↔ addOrderOf 1 ∣ n
Theorem: Ordinal.blsub_le_iff
File: Mathlib/SetTheory/Ordinal/Arithmetic.lean

tactic: convert bsup_le_iff.{_, v} (f := fun a ha => succ (f a ha)) (a := a) using 2
state_before: α : Type ?u.349869
β : Type ?u.349872
γ : Type ?u.349875
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal
f : (a : Ordinal) → a < o → Ordinal
a : Ordinal
⊢ blsub o f ≤ a ↔ ∀ (i : Ordinal) (h : i < o), f i h < a

tactic: simp_rw [succ_le_iff]
state_before: case h.e'_2.h.a
α : Type ?u.349869
β : Type ?u.349872
γ : Type ?u.349875
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal
f : (a : Ordinal) → a < o → Ordinal
a : Ordinal
a✝ : Ordinal
⊢ (∀ (h : a✝ < o), f a✝ h < a) ↔ ∀ (h : a✝ < o), succ (f a✝ h) ≤ a
Theorem: MeasureTheory.Measure.map_dirac
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

tactic: simp [hs, map_apply hf hs, hf hs, indicator_apply]
state_before: α : Type u_1
β : Type u_2
γ : Type ?u.329647
δ : Type ?u.329650
ι : Type ?u.329653
R : Type ?u.329656
R' : Type ?u.329659
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
inst✝ : MeasurableSpace α
f : α → β
hf : Measurable f
a : α
s : Set β
hs : MeasurableSet s
⊢ ↑↑(map f (dirac a)) s = ↑↑(dirac (f a)) s
Theorem: List.formPerm_apply_mem_eq_self_iff
File: Mathlib/GroupTheory/Perm/List.lean

tactic: obtain ⟨k, hk, rfl⟩ := nthLe_of_mem hx
state_before: α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : x ∈ l
⊢ ↑(formPerm l) x = x ↔ length l ≤ 1

tactic: rw [formPerm_apply_nthLe _ hl, hl.nthLe_inj_iff]
state_before: case intro.intro
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
⊢ ↑(formPerm l) (nthLe l k hk) = nthLe l k hk ↔ length l ≤ 1

tactic: cases hn : l.length
state_before: case intro.intro
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
⊢ (k + 1) % length l = k ↔ length l ≤ 1

tactic: exact absurd k.zero_le (hk.trans_le hn.le).not_le
state_before: case intro.intro.zero
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
hn : length l = Nat.zero
⊢ (k + 1) % Nat.zero = k ↔ Nat.zero ≤ 1

tactic: rw [hn] at hk
state_before: case intro.intro.succ
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
n✝ : ℕ
hn : length l = Nat.succ n✝
⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1

tactic: cases' (Nat.le_of_lt_succ hk).eq_or_lt with hk' hk'
state_before: case intro.intro.succ
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk✝ : k < length l
hx : nthLe l k hk✝ ∈ l
n✝ : ℕ
hk : k < Nat.succ n✝
hn : length l = Nat.succ n✝
⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1

tactic: simp [← hk', Nat.succ_le_succ_iff, eq_comm]
state_before: case intro.intro.succ.inl
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk✝ : k < length l
hx : nthLe l k hk✝ ∈ l
n✝ : ℕ
hk : k < Nat.succ n✝
hn : length l = Nat.succ n✝
hk' : k = n✝
⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1

tactic: simpa [Nat.mod_eq_of_lt (Nat.succ_lt_succ hk'), Nat.succ_lt_succ_iff] using
  k.zero_le.trans_lt hk'
state_before: case intro.intro.succ.inr
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk✝ : k < length l
hx : nthLe l k hk✝ ∈ l
n✝ : ℕ
hk : k < Nat.succ n✝
hn : length l = Nat.succ n✝
hk' : k < n✝
⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1
Theorem: Real.isLittleO_exp_comp_exp_comp
File: Mathlib/Analysis/SpecialFunctions/Exp.lean

tactic: simp only [isLittleO_iff_tendsto, exp_ne_zero, ← exp_sub, ← tendsto_neg_atTop_iff, false_imp_iff,
  imp_true_iff, tendsto_exp_comp_nhds_zero, neg_sub]
state_before: α : Type u_1
x y z : ℝ
l : Filter α
f g : α → ℝ
⊢ ((fun x => exp (f x)) =o[l] fun x => exp (g x)) ↔ Tendsto (fun x => g x - f x) l atTop
Theorem: MeasureTheory.integral_image_eq_integral_abs_deriv_smul
File: Mathlib/MeasureTheory/Function/Jacobian.lean

tactic: simpa only [det_one_smulRight] using
  integral_image_eq_integral_abs_det_fderiv_smul volume hs
    (fun x hx => (hf' x hx).hasFDerivWithinAt) hf g
state_before: E : Type ?u.1055778
F : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : IsAddHaarMeasure μ
s : Set ℝ
f f' : ℝ → ℝ
inst✝ : CompleteSpace F
hs : MeasurableSet s
hf' : ∀ (x : ℝ), x ∈ s → HasDerivWithinAt f (f' x) s x
hf : InjOn f s
g : ℝ → F
⊢ (∫ (x : ℝ) in f '' s, g x) = ∫ (x : ℝ) in s, abs (f' x) • g (f x)
Theorem: ZNum.add_one
File: Mathlib/Data/Num/Lemmas.lean

tactic: cases p <;> rfl
state_before: α : Type ?u.723974
p : PosNum
⊢ neg p + 1 = succ (neg p)
Theorem: MeasureTheory.Measure.map_toOuterMeasure
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

tactic: rw [← trimmed, OuterMeasure.trim_eq_trim_iff]
state_before: α : Type u_1
β : Type u_2
γ : Type ?u.213751
δ : Type ?u.213754
ι : Type ?u.213757
R : Type ?u.213760
R' : Type ?u.213763
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
hf : AEMeasurable f
⊢ ↑(map f μ) = OuterMeasure.trim (↑(OuterMeasure.map f) ↑μ)

tactic: intro s hs
state_before: α : Type u_1
β : Type u_2
γ : Type ?u.213751
δ : Type ?u.213754
ι : Type ?u.213757
R : Type ?u.213760
R' : Type ?u.213763
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
hf : AEMeasurable f
⊢ ∀ (s : Set β), MeasurableSet s → ↑↑(map f μ) s = ↑(↑(OuterMeasure.map f) ↑μ) s

tactic: rw [map_apply_of_aemeasurable hf hs, OuterMeasure.map_apply]
state_before: α : Type u_1
β : Type u_2
γ : Type ?u.213751
δ : Type ?u.213754
ι : Type ?u.213757
R : Type ?u.213760
R' : Type ?u.213763
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f : α → β
hf : AEMeasurable f
s : Set β
hs : MeasurableSet s
⊢ ↑↑(map f μ) s = ↑(↑(OuterMeasure.map f) ↑μ) s
Theorem: CategoryTheory.Limits.HasLimit.isoOfEquivalence_inv_π
File: Mathlib/CategoryTheory/Limits/HasLimits.lean

tactic: simp only [HasLimit.isoOfEquivalence, IsLimit.conePointsIsoOfEquivalence_hom]
state_before: J : Type u₁
inst✝⁴ : Category J
K : Type u₂
inst✝³ : Category K
C : Type u
inst✝² : Category C
F✝ F : J ⥤ C
inst✝¹ : HasLimit F
G : K ⥤ C
inst✝ : HasLimit G
e : J ≌ K
w : e.functor ⋙ G ≅ F
j : J
⊢ (isoOfEquivalence e w).inv ≫ limit.π F j = limit.π G (e.functor.obj j) ≫ w.hom.app j

tactic: dsimp
state_before: J : Type u₁
inst✝⁴ : Category J
K : Type u₂
inst✝³ : Category K
C : Type u
inst✝² : Category C
F✝ F : J ⥤ C
inst✝¹ : HasLimit F
G : K ⥤ C
inst✝ : HasLimit G
e : J ≌ K
w : e.functor ⋙ G ≅ F
j : J
⊢ (IsLimit.conePointsIsoOfEquivalence (limit.isLimit F) (limit.isLimit G) e w).inv ≫ limit.π F j =
    limit.π G (e.functor.obj j) ≫ w.hom.app j

tactic: simp
state_before: J : Type u₁
inst✝⁴ : Category J
K : Type u₂
inst✝³ : Category K
C : Type u
inst✝² : Category C
F✝ F : J ⥤ C
inst✝¹ : HasLimit F
G : K ⥤ C
inst✝ : HasLimit G
e : J ≌ K
w : e.functor ⋙ G ≅ F
j : J
⊢ limit.lift F ((Cones.postcompose w.hom).obj (Cone.whisker e.functor (limit.cone G))) ≫ limit.π F j =
    limit.π G (e.functor.obj j) ≫ w.hom.app j
Theorem: iteratedFDerivWithin_univ
File: Mathlib/Analysis/Calculus/ContDiffDef.lean

tactic: induction' n with n IH
state_before: 𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
m n✝ : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
⊢ iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n f

tactic: ext x
state_before: case zero
𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
⊢ iteratedFDerivWithin 𝕜 Nat.zero f univ = iteratedFDeriv 𝕜 Nat.zero f

tactic: simp
state_before: case zero.h.H
𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x✝¹ x₀ : E
c : F
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
x : E
x✝ : Fin Nat.zero → E
⊢ ↑(iteratedFDerivWithin 𝕜 Nat.zero f univ x) x✝ = ↑(iteratedFDeriv 𝕜 Nat.zero f x) x✝

tactic: ext (x m)
state_before: case succ
𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
m n✝ : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
IH : iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n f
⊢ iteratedFDerivWithin 𝕜 (Nat.succ n) f univ = iteratedFDeriv 𝕜 (Nat.succ n) f

tactic: rw [iteratedFDeriv_succ_apply_left, iteratedFDerivWithin_succ_apply_left, IH, fderivWithin_univ]
state_before: case succ.h.H
𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x✝ x₀ : E
c : F
m✝ n✝ : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
IH : iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n f
x : E
m : Fin (Nat.succ n) → E
⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.succ n) f univ x) m = ↑(iteratedFDeriv 𝕜 (Nat.succ n) f x) m
Theorem: CircleDeg1Lift.translate_zpow
File: Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean

tactic: simp only [← zsmul_eq_mul, ofAdd_zsmul, MonoidHom.map_zpow]
state_before: f g : CircleDeg1Lift
x : ℝ
n : ℤ
⊢ ↑translate (↑Multiplicative.ofAdd x) ^ n = ↑translate (↑Multiplicative.ofAdd (↑n * x))
Theorem: BoxIntegral.Prepartition.compl_congr
File: Mathlib/Analysis/BoxIntegral/Partition/Split.lean

tactic: dsimp only [compl]
state_before: ι : Type u_1
M : Type ?u.58340
n : ℕ
I J : Box ι
i : ι
x : ℝ
inst✝ : Finite ι
π₁ π₂ : Prepartition I
h : Prepartition.iUnion π₁ = Prepartition.iUnion π₂
⊢ compl π₁ = compl π₂

tactic: congr 1
state_before: ι : Type u_1
M : Type ?u.58340
n : ℕ
I J : Box ι
i : ι
x : ℝ
inst✝ : Finite ι
π₁ π₂ : Prepartition I
h : Prepartition.iUnion π₁ = Prepartition.iUnion π₂
⊢ Exists.choose (_ : ∃ π', Prepartition.iUnion π' = ↑I \ Prepartition.iUnion π₁) =
    Exists.choose (_ : ∃ π', Prepartition.iUnion π' = ↑I \ Prepartition.iUnion π₂)

tactic: rw [h]
state_before: case e_p
ι : Type u_1
M : Type ?u.58340
n : ℕ
I J : Box ι
i : ι
x : ℝ
inst✝ : Finite ι
π₁ π₂ : Prepartition I
h : Prepartition.iUnion π₁ = Prepartition.iUnion π₂
⊢ (fun π' => Prepartition.iUnion π' = ↑I \ Prepartition.iUnion π₁) = fun π' =>
    Prepartition.iUnion π' = ↑I \ Prepartition.iUnion π₂
Theorem: ENNReal.iUnion_Ioc_coe_nat
File: Mathlib/Data/Real/ENNReal.lean

tactic: simp only [← Ioi_inter_Iic, ← inter_iUnion, iUnion_Iic_coe_nat, diff_eq]
state_before: α : Type ?u.133965
β : Type ?u.133968
a b c d : ℝ≥0∞
r p q : ℝ≥0
⊢ (⋃ (n : ℕ), Ioc a ↑n) = Ioi a \ {⊤}
Theorem: Finset.not_mem_sdiff_of_not_mem_left
File: Mathlib/Data/Finset/Basic.lean

tactic: simp [h]
state_before: α : Type u_1
β : Type ?u.223921
γ : Type ?u.223924
inst✝ : DecidableEq α
s t u v : Finset α
a b : α
h : ¬a ∈ s
⊢ ¬a ∈ s \ t
Theorem: Metric.mem_cthickening_of_dist_le
File: Mathlib/Topology/MetricSpace/HausdorffDistance.lean

tactic: apply mem_cthickening_of_edist_le x y δ E h
state_before: ι : Sort ?u.98897
α✝ : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α✝
δ✝ ε : ℝ
s t : Set α✝
x✝ : α✝
α : Type u_1
inst✝ : PseudoMetricSpace α
x y : α
δ : ℝ
E : Set α
h : y ∈ E
h' : dist x y ≤ δ
⊢ x ∈ cthickening δ E

tactic: rw [edist_dist]
state_before: ι : Sort ?u.98897
α✝ : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α✝
δ✝ ε : ℝ
s t : Set α✝
x✝ : α✝
α : Type u_1
inst✝ : PseudoMetricSpace α
x y : α
δ : ℝ
E : Set α
h : y ∈ E
h' : dist x y ≤ δ
⊢ edist x y ≤ ENNReal.ofReal δ

tactic: exact ENNReal.ofReal_le_ofReal h'
state_before: ι : Sort ?u.98897
α✝ : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α✝
δ✝ ε : ℝ
s t : Set α✝
x✝ : α✝
α : Type u_1
inst✝ : PseudoMetricSpace α
x y : α
δ : ℝ
E : Set α
h : y ∈ E
h' : dist x y ≤ δ
⊢ ENNReal.ofReal (dist x y) ≤ ENNReal.ofReal δ
Theorem: BilinForm.mul_toMatrix_mul
File: Mathlib/LinearAlgebra/Matrix/BilinearForm.lean

tactic: simp only [B.toMatrix_comp b c, toMatrix_toLin, transpose_transpose]
state_before: R : Type ?u.1640706
M✝ : Type ?u.1640709
inst✝²⁰ : Semiring R
inst✝¹⁹ : AddCommMonoid M✝
inst✝¹⁸ : Module R M✝
R₁ : Type ?u.1640745
M₁ : Type ?u.1640748
inst✝¹⁷ : Ring R₁
inst✝¹⁶ : AddCommGroup M₁
inst✝¹⁵ : Module R₁ M₁
R₂ : Type u_1
M₂ : Type u_2
inst✝¹⁴ : CommSemiring R₂
inst✝¹³ : AddCommMonoid M₂
inst✝¹² : Module R₂ M₂
R₃ : Type ?u.1641547
M₃ : Type ?u.1641550
inst✝¹¹ : CommRing R₃
inst✝¹⁰ : AddCommGroup M₃
inst✝⁹ : Module R₃ M₃
V : Type ?u.1642138
K : Type ?u.1642141
inst✝⁸ : Field K
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module K V
B✝ : BilinForm R M✝
B₁ : BilinForm R₁ M₁
B₂ : BilinForm R₂ M₂
n : Type u_4
o : Type u_3
inst✝⁵ : Fintype n
inst✝⁴ : Fintype o
inst✝³ : DecidableEq n
b : Basis n R₂ M₂
M₂' : Type u_5
inst✝² : AddCommMonoid M₂'
inst✝¹ : Module R₂ M₂'
c : Basis o R₂ M₂'
inst✝ : DecidableEq o
B : BilinForm R₂ M₂
M : Matrix o n R₂
N : Matrix n o R₂
⊢ M ⬝ ↑(toMatrix b) B ⬝ N = ↑(toMatrix c) (comp B (↑(Matrix.toLin c b) Mᵀ) (↑(Matrix.toLin c b) N))