1337xyz1337xyz/leandojo-benchmark4-v10 huggingface dataset:
Theorem: PowerSeries.order_zero_of_unit
File: Mathlib/RingTheory/PowerSeries/Order.lean

tactic: rintro ⟨⟨u, v, hu, hv⟩, hf⟩
state_before: R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f : R⟦X⟧
⊢ IsUnit f → f.order = 0
state_after: case intro.mk
R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f u v : R⟦X⟧
hu : u * v = 1
hv : v * u = 1
hf : ↑{ val := u, inv := v, val_inv := hu, inv_val := hv } = f
⊢ f.order = 0

tactic: apply And.left
state_before: case intro.mk
R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f u v : R⟦X⟧
hu : u * v = 1
hv : v * u = 1
hf : ↑{ val := u, inv := v, val_inv := hu, inv_val := hv } = f
⊢ f.order = 0
state_after: case intro.mk.self
R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f u v : R⟦X⟧
hu : u * v = 1
hv : v * u = 1
hf : ↑{ val := u, inv := v, val_inv := hu, inv_val := hv } = f
⊢ f.order = 0 ∧ ?intro.mk.b

case intro.mk.b
R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f u v : R⟦X⟧
hu : u * v = 1
hv : v * u = 1
hf : ↑{ val := u, inv := v, val_inv := hu, inv_val := hv } = f
⊢ Prop

tactic: rw [← add_eq_zero_iff, ← hf, ← nonpos_iff_eq_zero, ← @order_one R _ _, ← hu]
state_before: case intro.mk.self
R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f u v : R⟦X⟧
hu : u * v = 1
hv : v * u = 1
hf : ↑{ val := u, inv := v, val_inv := hu, inv_val := hv } = f
⊢ f.order = 0 ∧ ?intro.mk.b

case intro.mk.b
R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f u v : R⟦X⟧
hu : u * v = 1
hv : v * u = 1
hf : ↑{ val := u, inv := v, val_inv := hu, inv_val := hv } = f
⊢ Prop
state_after: case intro.mk.self
R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f u v : R⟦X⟧
hu : u * v = 1
hv : v * u = 1
hf : ↑{ val := u, inv := v, val_inv := hu, inv_val := hv } = f
⊢ (↑{ val := u, inv := v, val_inv := hu, inv_val := hv }).order + ?m.107073 ≤ (u * v).order

R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f u v : R⟦X⟧
hu : u * v = 1
hv : v * u = 1
hf : ↑{ val := u, inv := v, val_inv := hu, inv_val := hv } = f
⊢ PartENat

R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f u v : R⟦X⟧
hu : u * v = 1
hv : v * u = 1
hf : ↑{ val := u, inv := v, val_inv := hu, inv_val := hv } = f
⊢ PartENat

tactic: exact order_mul_ge _ _
state_before: case intro.mk.self
R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f u v : R⟦X⟧
hu : u * v = 1
hv : v * u = 1
hf : ↑{ val := u, inv := v, val_inv := hu, inv_val := hv } = f
⊢ (↑{ val := u, inv := v, val_inv := hu, inv_val := hv }).order + ?m.107073 ≤ (u * v).order

R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f u v : R⟦X⟧
hu : u * v = 1
hv : v * u = 1
hf : ↑{ val := u, inv := v, val_inv := hu, inv_val := hv } = f
⊢ PartENat

R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
f u v : R⟦X⟧
hu : u * v = 1
hv : v * u = 1
hf : ↑{ val := u, inv := v, val_inv := hu, inv_val := hv } = f
⊢ PartENat
state_after: no goals
Theorem: Polynomial.X_sub_C_pow_dvd_iff
File: Mathlib/Algebra/Polynomial/RingDivision.lean

tactic: convert (map_dvd_iff <| algEquivAevalXAddC t).symm using 2
state_before: R : Type u
S : Type v
T : Type w
a b : R
n✝ : ℕ
inst✝ : CommRing R
p : R[X]
t : R
n : ℕ
⊢ (X - C t) ^ n ∣ p ↔ X ^ n ∣ p.comp (X + C t)
state_after: case h.e'_2.h.e'_3
R : Type u
S : Type v
T : Type w
a b : R
n✝ : ℕ
inst✝ : CommRing R
p : R[X]
t : R
n : ℕ
⊢ X ^ n = (algEquivAevalXAddC t) ((X - C t) ^ n)

tactic: simp [C_eq_algebraMap]
state_before: case h.e'_2.h.e'_3
R : Type u
S : Type v
T : Type w
a b : R
n✝ : ℕ
inst✝ : CommRing R
p : R[X]
t : R
n : ℕ
⊢ X ^ n = (algEquivAevalXAddC t) ((X - C t) ^ n)
state_after: no goals
Theorem: MeasureTheory.Measure.integrable_compProd_iff
File: Mathlib/Probability/Kernel/MeasureCompProd.lean

tactic: rw [Measure.compProd, ProbabilityTheory.integrable_compProd_iff hf]
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : ↥(kernel α β)
inst✝² : SFinite μ
inst✝¹ : IsSFiniteKernel κ
E : Type u_3
inst✝ : NormedAddCommGroup E
f : α × β → E
hf : AEStronglyMeasurable f (μ ⊗ₘ κ)
⊢ Integrable f (μ ⊗ₘ κ) ↔
    (∀ᵐ (x : α) ∂μ, Integrable (fun y => f (x, y)) (κ x)) ∧ Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂κ x) μ
state_after: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : ↥(kernel α β)
inst✝² : SFinite μ
inst✝¹ : IsSFiniteKernel κ
E : Type u_3
inst✝ : NormedAddCommGroup E
f : α × β → E
hf : AEStronglyMeasurable f (μ ⊗ₘ κ)
⊢ (∀ᵐ (x : α) ∂(kernel.const Unit μ) (), Integrable (fun y => f (x, y)) ((kernel.prodMkLeft Unit κ) ((), x))) ∧
      Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂(kernel.prodMkLeft Unit κ) ((), x)) ((kernel.const Unit μ) ()) ↔
    (∀ᵐ (x : α) ∂μ, Integrable (fun y => f (x, y)) (κ x)) ∧ Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂κ x) μ

tactic: rfl
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : ↥(kernel α β)
inst✝² : SFinite μ
inst✝¹ : IsSFiniteKernel κ
E : Type u_3
inst✝ : NormedAddCommGroup E
f : α × β → E
hf : AEStronglyMeasurable f (μ ⊗ₘ κ)
⊢ (∀ᵐ (x : α) ∂(kernel.const Unit μ) (), Integrable (fun y => f (x, y)) ((kernel.prodMkLeft Unit κ) ((), x))) ∧
      Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂(kernel.prodMkLeft Unit κ) ((), x)) ((kernel.const Unit μ) ()) ↔
    (∀ᵐ (x : α) ∂μ, Integrable (fun y => f (x, y)) (κ x)) ∧ Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂κ x) μ
state_after: no goals
Theorem: CategoryTheory.NonPreadditiveAbelian.sub_zero
File: Mathlib/CategoryTheory/Abelian/NonPreadditive.lean

tactic: rw [sub_def]
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
⊢ a - 0 = a
state_after: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
⊢ prod.lift a 0 ≫ σ = a

tactic: conv_lhs =>
  congr; congr; rw [← Category.comp_id a]
  case a.g => rw [show 0 = a ≫ (0 : Y ⟶ Y) by simp]
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
⊢ prod.lift a 0 ≫ σ = a
state_after: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
⊢ prod.lift (a ≫ 𝟙 Y) (a ≫ 0) ≫ σ = a

tactic: rw [← prod.comp_lift, Category.assoc, lift_σ, Category.comp_id]
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
⊢ prod.lift (a ≫ 𝟙 Y) (a ≫ 0) ≫ σ = a
state_after: no goals

tactic: simp
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
⊢ 0 = a ≫ 0
state_after: no goals
Theorem: CategoryTheory.MonoidalCategory.comp_tensor_id
File: Mathlib/CategoryTheory/Monoidal/Category.lean

tactic: simp
state_before: C✝ : Type u
𝒞 : Category.{v, u} C✝
inst✝² : MonoidalCategory C✝
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : MonoidalCategory C
U V W X Y Z : C
f : W ⟶ X
g : X ⟶ Y
⊢ f ≫ g ⊗ 𝟙 Z = (f ⊗ 𝟙 Z) ≫ (g ⊗ 𝟙 Z)
state_after: no goals
Theorem: Equiv.Perm.sign_inv
File: Mathlib/GroupTheory/Perm/Sign.lean

tactic: rw [MonoidHom.map_inv sign f, Int.units_inv_eq_self]
state_before: α : Type u
inst✝¹ : DecidableEq α
β : Type v
inst✝ : Fintype α
f : Perm α
⊢ sign f⁻¹ = sign f
state_after: no goals
Theorem: MeasureTheory.tilted_apply
File: Mathlib/MeasureTheory/Measure/Tilted.lean

tactic: rw [Measure.tilted, withDensity_apply' _ s]
state_before: α : Type u_1
mα : MeasurableSpace α
μ✝ : Measure α
f✝ : α → ℝ
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
s : Set α
⊢ (μ.tilted f) s = ∫⁻ (a : α) in s, ENNReal.ofReal (rexp (f a) / ∫ (x : α), rexp (f x) ∂μ) ∂μ
state_after: no goals
Theorem: AlgebraicGeometry.Spec.locallyRingedSpaceMap_comp
File: Mathlib/AlgebraicGeometry/Spec.lean

tactic: rw [Spec.locallyRingedSpaceMap_val, Spec.sheafedSpaceMap_comp]
state_before: R S T : CommRingCat
f : R ⟶ S
g : S ⟶ T
⊢ (locallyRingedSpaceMap (f ≫ g)).val = (locallyRingedSpaceMap g ≫ locallyRingedSpaceMap f).val
state_after: R S T : CommRingCat
f : R ⟶ S
g : S ⟶ T
⊢ sheafedSpaceMap g ≫ sheafedSpaceMap f = (locallyRingedSpaceMap g ≫ locallyRingedSpaceMap f).val

tactic: rfl
state_before: R S T : CommRingCat
f : R ⟶ S
g : S ⟶ T
⊢ sheafedSpaceMap g ≫ sheafedSpaceMap f = (locallyRingedSpaceMap g ≫ locallyRingedSpaceMap f).val
state_after: no goals
Theorem: IntermediateField.neg_mem
File: Mathlib/FieldTheory/IntermediateField.lean

tactic: show -x ∈S.toSubalgebra
state_before: K : Type u_1
L : Type u_2
L' : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Field L'
inst✝¹ : Algebra K L
inst✝ : Algebra K L'
S : IntermediateField K L
x : L
hx : x ∈ S
⊢ -x ∈ S
state_after: K : Type u_1
L : Type u_2
L' : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Field L'
inst✝¹ : Algebra K L
inst✝ : Algebra K L'
S : IntermediateField K L
x : L
hx : x ∈ S
⊢ -x ∈ S.toSubalgebra

tactic: simpa
state_before: K : Type u_1
L : Type u_2
L' : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Field L'
inst✝¹ : Algebra K L
inst✝ : Algebra K L'
S : IntermediateField K L
x : L
hx : x ∈ S
⊢ -x ∈ S.toSubalgebra
state_after: no goals
Theorem: NumberField.InfinitePlace.embedding_mk_eq_of_isReal
File: Mathlib/NumberTheory/NumberField/Embeddings.lean

tactic: have := embedding_mk_eq φ
state_before: k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
F : Type u_3
inst✝ : Field F
φ : K →+* ℂ
h : ComplexEmbedding.IsReal φ
⊢ (mk φ).embedding = φ
state_after: k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
F : Type u_3
inst✝ : Field F
φ : K →+* ℂ
h : ComplexEmbedding.IsReal φ
this : (mk φ).embedding = φ ∨ (mk φ).embedding = ComplexEmbedding.conjugate φ
⊢ (mk φ).embedding = φ

tactic: rwa [ComplexEmbedding.isReal_iff.mp h, or_self] at this
state_before: k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
F : Type u_3
inst✝ : Field F
φ : K →+* ℂ
h : ComplexEmbedding.IsReal φ
this : (mk φ).embedding = φ ∨ (mk φ).embedding = ComplexEmbedding.conjugate φ
⊢ (mk φ).embedding = φ
state_after: no goals
Theorem: inf_idem
File: Mathlib/Order/Lattice.lean

tactic: simp
state_before: α : Type u
β : Type v
inst✝ : SemilatticeInf α
a✝ b c d a : α
⊢ a ⊓ a = a
state_after: no goals
Theorem: Finset.offDiag_union
File: Mathlib/Data/Finset/Prod.lean

tactic: push_cast
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s t : Finset α
x : α × α
h : _root_.Disjoint s t
⊢ ↑(s ∪ t).offDiag = ↑(s.offDiag ∪ t.offDiag ∪ s ×ˢ t ∪ t ×ˢ s)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s t : Finset α
x : α × α
h : _root_.Disjoint s t
⊢ (↑s ∪ ↑t).offDiag = (↑s).offDiag ∪ (↑t).offDiag ∪ ↑s ×ˢ ↑t ∪ ↑t ×ˢ ↑s

tactic: exact Set.offDiag_union (disjoint_coe.2 h)
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s t : Finset α
x : α × α
h : _root_.Disjoint s t
⊢ (↑s ∪ ↑t).offDiag = (↑s).offDiag ∪ (↑t).offDiag ∪ ↑s ×ˢ ↑t ∪ ↑t ×ˢ ↑s
state_after: no goals
Theorem: LieHom.range_eq_map
File: Mathlib/Algebra/Lie/Subalgebra.lean

tactic: ext
state_before: R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
K K' : LieSubalgebra R L
K₂ : LieSubalgebra R L₂
⊢ f.range = map f ⊤
state_after: case h
R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
K K' : LieSubalgebra R L
K₂ : LieSubalgebra R L₂
x✝ : L₂
⊢ x✝ ∈ f.range ↔ x✝ ∈ map f ⊤

tactic: simp
state_before: case h
R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
K K' : LieSubalgebra R L
K₂ : LieSubalgebra R L₂
x✝ : L₂
⊢ x✝ ∈ f.range ↔ x✝ ∈ map f ⊤
state_after: no goals
Theorem: Equiv.naturality
File: Mathlib/Control/Traversable/Equiv.lean

tactic: simp only [Equiv.traverse, functor_norm]
state_before: t t' : Type u → Type u
eqv : (α : Type u) → t α ≃ t' α
inst✝⁵ : Traversable t
inst✝⁴ : LawfulTraversable t
F G : Type u → Type u
inst✝³ : Applicative F
inst✝² : Applicative G
inst✝¹ : LawfulApplicative F
inst✝ : LawfulApplicative G
η : ApplicativeTransformation F G
α β γ : Type u
f : α → F β
x : t' α
⊢ (fun {α} => η.app α) (Equiv.traverse eqv f x) = Equiv.traverse eqv ((fun {α} => η.app α) ∘ f) x
state_after: no goals
Theorem: bernsteinPolynomial.iterate_derivative_at_0_ne_zero
File: Mathlib/RingTheory/Polynomial/Bernstein.lean

tactic: simp only [Int.natCast_eq_zero, bernsteinPolynomial.iterate_derivative_at_0, Ne, Nat.cast_eq_zero]
state_before: R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
⊢ eval 0 ((⇑derivative)^[ν] (bernsteinPolynomial R n ν)) ≠ 0
state_after: R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
⊢ ¬eval (↑(n - (ν - 1))) (ascPochhammer R ν) = 0

tactic: simp only [← ascPochhammer_eval_cast]
state_before: R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
⊢ ¬eval (↑(n - (ν - 1))) (ascPochhammer R ν) = 0
state_after: R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
⊢ ¬↑(eval (n - (ν - 1)) (ascPochhammer ℕ ν)) = 0

tactic: norm_cast
state_before: R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
⊢ ¬↑(eval (n - (ν - 1)) (ascPochhammer ℕ ν)) = 0
state_after: R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
⊢ ¬eval (n - (ν - 1)) (ascPochhammer ℕ ν) = 0

tactic: apply ne_of_gt
state_before: R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
⊢ ¬eval (n - (ν - 1)) (ascPochhammer ℕ ν) = 0
state_after: case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
⊢ 0 < eval (n - (ν - 1)) (ascPochhammer ℕ ν)

tactic: obtain rfl | h' := Nat.eq_zero_or_pos ν
state_before: case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
⊢ 0 < eval (n - (ν - 1)) (ascPochhammer ℕ ν)
state_after: case h.inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n : ℕ
h : 0 ≤ n
⊢ 0 < eval (n - (0 - 1)) (ascPochhammer ℕ 0)

case h.inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
h' : ν > 0
⊢ 0 < eval (n - (ν - 1)) (ascPochhammer ℕ ν)

tactic: simp
state_before: case h.inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n : ℕ
h : 0 ≤ n
⊢ 0 < eval (n - (0 - 1)) (ascPochhammer ℕ 0)
state_after: no goals

tactic: rw [← Nat.succ_pred_eq_of_pos h'] at h
state_before: case h.inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν ≤ n
h' : ν > 0
⊢ 0 < eval (n - (ν - 1)) (ascPochhammer ℕ ν)
state_after: case h.inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν.pred.succ ≤ n
h' : ν > 0
⊢ 0 < eval (n - (ν - 1)) (ascPochhammer ℕ ν)

tactic: exact ascPochhammer_pos _ _ (tsub_pos_of_lt (Nat.lt_of_succ_le h))
state_before: case h.inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n ν : ℕ
h : ν.pred.succ ≤ n
h' : ν > 0
⊢ 0 < eval (n - (ν - 1)) (ascPochhammer ℕ ν)
state_after: no goals
Theorem: iteratedDeriv_succ
File: Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean

tactic: ext x
state_before: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
n : ℕ
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
⊢ iteratedDeriv (n + 1) f = deriv (iteratedDeriv n f)
state_after: case h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
n : ℕ
f : 𝕜 → F
s : Set 𝕜
x✝ x : 𝕜
⊢ iteratedDeriv (n + 1) f x = deriv (iteratedDeriv n f) x

tactic: rw [← iteratedDerivWithin_univ, ← iteratedDerivWithin_univ, ← derivWithin_univ]
state_before: case h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
n : ℕ
f : 𝕜 → F
s : Set 𝕜
x✝ x : 𝕜
⊢ iteratedDeriv (n + 1) f x = deriv (iteratedDeriv n f) x
state_after: case h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
n : ℕ
f : 𝕜 → F
s : Set 𝕜
x✝ x : 𝕜
⊢ iteratedDerivWithin (n + 1) f univ x = derivWithin (iteratedDerivWithin n f univ) univ x

tactic: exact iteratedDerivWithin_succ uniqueDiffWithinAt_univ
state_before: case h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
n : ℕ
f : 𝕜 → F
s : Set 𝕜
x✝ x : 𝕜
⊢ iteratedDerivWithin (n + 1) f univ x = derivWithin (iteratedDerivWithin n f univ) univ x
state_after: no goals
Theorem: MeasureTheory.Lp.norm_const'
File: Mathlib/MeasureTheory/Function/LpSpace.lean

tactic: rw [← Memℒp.toLp_const, Lp.norm_toLp, snorm_const'] <;> try assumption
state_before: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : IsFiniteMeasure μ
c : E
hp_zero : p ≠ 0
hp_top : p ≠ ⊤
⊢ ‖(Lp.const p μ) c‖ = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)
state_after: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : IsFiniteMeasure μ
c : E
hp_zero : p ≠ 0
hp_top : p ≠ ⊤
⊢ (↑‖c‖₊ * μ Set.univ ^ (1 / p.toReal)).toReal = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)

tactic: rw [ENNReal.toReal_mul, ENNReal.coe_toReal, ← ENNReal.toReal_rpow, coe_nnnorm]
state_before: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : IsFiniteMeasure μ
c : E
hp_zero : p ≠ 0
hp_top : p ≠ ⊤
⊢ (↑‖c‖₊ * μ Set.univ ^ (1 / p.toReal)).toReal = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)
state_after: no goals

tactic: assumption
state_before: case h_top
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : IsFiniteMeasure μ
c : E
hp_zero : p ≠ 0
hp_top : p ≠ ⊤
⊢ p ≠ ⊤
state_after: no goals
Theorem: hasFPowerSeriesOnBall_const
File: Mathlib/Analysis/Analytic/Basic.lean

tactic: refine ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => ?_⟩
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r r' : ℝ≥0∞
c : F
e : E
⊢ HasFPowerSeriesOnBall (fun x => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r r' : ℝ≥0∞
c : F
e y✝ : E
x✝ : y✝ ∈ EMetric.ball 0 ⊤
n : ℕ
hn : n ≠ 0
⊢ ((constFormalMultilinearSeries 𝕜 E c n) fun x => y✝) = 0

tactic: simp [constFormalMultilinearSeries_apply hn]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r r' : ℝ≥0∞
c : F
e y✝ : E
x✝ : y✝ ∈ EMetric.ball 0 ⊤
n : ℕ
hn : n ≠ 0
⊢ ((constFormalMultilinearSeries 𝕜 E c n) fun x => y✝) = 0
state_after: no goals

tactic: simp
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r r' : ℝ≥0∞
c : F
e : E
⊢ ⊤ ≤ (constFormalMultilinearSeries 𝕜 E c).radius
state_after: no goals
Theorem: NonUnitalRingHom.prod_unique
File: Mathlib/Algebra/Ring/Prod.lean

tactic: simp only [prod_apply, coe_fst, coe_snd, comp_apply]
state_before: α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
S : Type u_5
S' : Type u_6
T : Type u_7
T' : Type u_8
inst✝² : NonUnitalNonAssocSemiring R
inst✝¹ : NonUnitalNonAssocSemiring S
inst✝ : NonUnitalNonAssocSemiring T
f✝ : R →ₙ+* S
g : R →ₙ+* T
f : R →ₙ+* S × T
x : R
⊢ (((fst S T).comp f).prod ((snd S T).comp f)) x = f x
state_after: no goals
Theorem: Orientation.oangle_self
File: Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean

tactic: rw [oangle, kahler_apply_self, ← ofReal_pow]
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 : V
⊢ o.oangle x x = 0
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 : V
⊢ ↑(↑(‖x‖ ^ 2)).arg = 0

tactic: convert QuotientAddGroup.mk_zero (AddSubgroup.zmultiples (2 * π))
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 : V
⊢ ↑(↑(‖x‖ ^ 2)).arg = 0
state_after: case h.e'_2.h.h.e'_1
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 : V
e_1✝ : Real.Angle = (ℝ ⧸ AddSubgroup.zmultiples (2 * π))
⊢ (↑(‖x‖ ^ 2)).arg = 0

tactic: apply arg_ofReal_of_nonneg
state_before: case h.e'_2.h.h.e'_1
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 : V
e_1✝ : Real.Angle = (ℝ ⧸ AddSubgroup.zmultiples (2 * π))
⊢ (↑(‖x‖ ^ 2)).arg = 0
state_after: case h.e'_2.h.h.e'_1.hx
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 : V
e_1✝ : Real.Angle = (ℝ ⧸ AddSubgroup.zmultiples (2 * π))
⊢ 0 ≤ ‖x‖ ^ 2

tactic: positivity
state_before: case h.e'_2.h.h.e'_1.hx
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 : V
e_1✝ : Real.Angle = (ℝ ⧸ AddSubgroup.zmultiples (2 * π))
⊢ 0 ≤ ‖x‖ ^ 2
state_after: no goals
Theorem: Sum.getLeft_eq_getLeft?
File: Mathlib/Data/Sum/Basic.lean

tactic: simp [← getLeft?_eq_some_iff]
state_before: α : Type u
α' : Type w
β : Type v
β' : Type x
γ : Type u_1
δ : Type u_2
x y : α ⊕ β
h₁ : x.isLeft = true
h₂ : x.getLeft?.isSome = true
⊢ x.getLeft h₁ = x.getLeft?.get h₂
state_after: no goals