ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: UniformSpace.to_nhds_mono
File: Mathlib/Topology/UniformSpace/Basic.lean

Proof steps:
state_before: α : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
u₁ u₂ : UniformSpace α
h : u₁ ≤ u₂
a : α
⊢ 𝓝 a ≤ 𝓝 a
state_after: α : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
u₁ u₂ : UniformSpace α
h : u₁ ≤ u₂
a : α
⊢ (𝓤 α).lift' (ball a) ≤ (𝓤 α).lift' (ball a)
tactic: rw [@nhds_eq_uniformity α u₁ a, @nhds_eq_uniformity α u₂ a]
state_before: α : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
u₁ u₂ : UniformSpace α
h : u₁ ≤ u₂
a : α
⊢ (𝓤 α).lift' (ball a) ≤ (𝓤 α).lift' (ball a)
state_after: no goals
tactic: exact lift'_mono h le_rfl
Theorem: LieAlgebra.IsKilling.traceForm_cartan_nondegenerate
File: Mathlib/Algebra/Lie/Weights/Killing.lean

Proof steps:
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : Field K
inst✝⁶ : LieAlgebra K L
inst✝⁵ : IsKilling R L
inst✝⁴ : Module.Finite R L
inst✝³ : Module.Free R L
inst✝² : IsNoetherian R L
inst✝¹ : IsArtinian R L
H : LieSubalgebra R L
inst✝ : H.IsCartanSubalgebra
⊢ (LieModule.traceForm R (↥H) L).Nondegenerate
state_after: no goals
tactic: simp [LinearMap.BilinForm.nondegenerate_iff_ker_eq_bot]
Theorem: CochainComplex.HomComplex.Cochain.leftShift_units_smul
File: Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean

Proof steps:
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n : ℤ
γ γ₁ γ₂ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
x : Rˣ
⊢ (x • γ).leftShift a n' hn' = x • γ.leftShift a n' hn'
state_after: no goals
tactic: apply leftShift_smul
Theorem: mem_convexHull_iff
File: Mathlib/Analysis/Convex/Hull.lean

Proof steps:
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s t : Set E
x y : E
⊢ x ∈ (convexHull 𝕜) s ↔ ∀ (t : Set E), s ⊆ t → Convex 𝕜 t → x ∈ t
state_after: no goals
tactic: simp_rw [convexHull_eq_iInter, mem_iInter]
Theorem: Int.not_isField
File: Mathlib/NumberTheory/NumberField/Basic.lean

Proof steps:
state_before: h : IsField ℤ
a : ℤ
⊢ 2 * a = 1 → 1 = a + a
state_after: h : IsField ℤ
a : ℤ
⊢ 2 * a = 1 → 1 = 2 * a
tactic: rw [← two_mul]
state_before: h : IsField ℤ
a : ℤ
⊢ 2 * a = 1 → 1 = 2 * a
state_after: no goals
tactic: exact Eq.symm
Theorem: Cubic.toPoly_eq_zero_iff
File: Mathlib/Algebra/CubicDiscriminant.lean

Proof steps:
state_before: R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P✝ Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
P : Cubic R
⊢ P.toPoly = 0 ↔ P = 0
state_after: no goals
tactic: rw [← zero, toPoly_injective]
Theorem: BoxIntegral.Box.splitLower_le
File: Mathlib/Analysis/BoxIntegral/Partition/Split.lean

Proof steps:
state_before: ι : Type u_1
M : Type u_2
n : ℕ
I : Box ι
i : ι
x : ℝ
y : ι → ℝ
⊢ ↑(I.splitLower i x) ⊆ ↑↑I
state_after: no goals
tactic: simp
Theorem: Orientation.map_eq_iff_det_pos
File: Mathlib/LinearAlgebra/Orientation.lean

Proof steps:
state_before: R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
⊢ (map ι f) x = x ↔ 0 < LinearMap.det ↑f
state_after: case inl
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
⊢ (map ι f) x = x ↔ 0 < LinearMap.det ↑f

case inr
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
⊢ (map ι f) x = x ↔ 0 < LinearMap.det ↑f
tactic: cases isEmpty_or_nonempty ι
state_before: case inr
R : Type u_1
inst✝³ : LinearOrderedField R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
_i : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
⊢ (map ι f) x = x ↔ 0 < LinearMap.det ↑f
state_after: no goals
tactic: rw [map_eq_det_inv_smul _ _ h, units_inv_smul, units_smul_eq_self_iff, LinearEquiv.coe_det]
Theorem: Real.sSup_smul_of_nonneg
File: Mathlib/Data/Real/Pointwise.lean

Proof steps:
state_before: ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
⊢ sSup (a • s) = a • sSup s
state_after: case inl
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
⊢ sSup (a • ∅) = a • sSup ∅

case inr
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
⊢ sSup (a • s) = a • sSup s
tactic: obtain rfl | hs := s.eq_empty_or_nonempty
state_before: case inr
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
⊢ sSup (a • s) = a • sSup s
state_after: case inr.inl
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
s : Set ℝ
hs : s.Nonempty
ha : 0 ≤ 0
⊢ sSup (0 • s) = 0 • sSup s

case inr.inr
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
⊢ sSup (a • s) = a • sSup s
tactic: obtain rfl | ha' := ha.eq_or_lt
state_before: case inr.inr
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
⊢ sSup (a • s) = a • sSup s
state_after: case pos
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
h : BddAbove s
⊢ sSup (a • s) = a • sSup s

case neg
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
h : ¬BddAbove s
⊢ sSup (a • s) = a • sSup s
tactic: by_cases h : BddAbove s
Theorem: LieSubmodule.lowerCentralSeries_tensor_eq_baseChange
File: Mathlib/Algebra/Lie/Nilpotent.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
k : ℕ
⊢ lowerCentralSeries A (A ⊗[R] L) (A ⊗[R] M) k = baseChange A (lowerCentralSeries R L M k)
state_after: case zero
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
⊢ lowerCentralSeries A (A ⊗[R] L) (A ⊗[R] M) 0 = baseChange A (lowerCentralSeries R L M 0)

case succ
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
k : ℕ
ih : lowerCentralSeries A (A ⊗[R] L) (A ⊗[R] M) k = baseChange A (lowerCentralSeries R L M k)
⊢ lowerCentralSeries A (A ⊗[R] L) (A ⊗[R] M) (k + 1) = baseChange A (lowerCentralSeries R L M (k + 1))
tactic: induction' k with k ih
state_before: case succ
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
k : ℕ
ih : lowerCentralSeries A (A ⊗[R] L) (A ⊗[R] M) k = baseChange A (lowerCentralSeries R L M k)
⊢ lowerCentralSeries A (A ⊗[R] L) (A ⊗[R] M) (k + 1) = baseChange A (lowerCentralSeries R L M (k + 1))
state_after: no goals
tactic: simp only [lowerCentralSeries_succ, ih, ← baseChange_top, lie_baseChange]
Theorem: Mathlib.Tactic.RingNF.nat_rawCast_1
File: Mathlib/Tactic/Ring/RingNF.lean

Proof steps:
state_before: R : Type u_1
inst✝ : CommSemiring R
n d : ℕ
⊢ Nat.rawCast 1 = 1
state_after: no goals
tactic: simp
Theorem: Cardinal.two_le_iff'
File: Mathlib/SetTheory/Cardinal/Basic.lean

Proof steps:
state_before: α β : Type u
c : Cardinal.{?u.233529}
x : α
⊢ 2 ≤ #α ↔ ∃ y, y ≠ x
state_after: no goals
tactic: rw [two_le_iff, ← nontrivial_iff, nontrivial_iff_exists_ne x]
Theorem: ComputablePred.to_re
File: Mathlib/Computability/Halting.lean

Proof steps:
state_before: α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
p : α → Prop
hp : ComputablePred p
⊢ RePred p
state_after: case intro.intro
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → Bool
hf : Computable f
hp : ComputablePred fun a => f a = true
⊢ RePred fun a => f a = true
tactic: obtain ⟨f, hf, rfl⟩ := computable_iff.1 hp
state_before: case intro.intro
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → Bool
hf : Computable f
hp : ComputablePred fun a => f a = true
⊢ RePred fun a => f a = true
state_after: case intro.intro
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → Bool
hf : Computable f
hp : ComputablePred fun a => f a = true
⊢ Partrec fun a => Part.assert ((fun a => f a = true) a) fun x => Part.some ()
tactic: unfold RePred
state_before: case intro.intro
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → Bool
hf : Computable f
hp : ComputablePred fun a => f a = true
⊢ Partrec fun a => Part.assert ((fun a => f a = true) a) fun x => Part.some ()
state_after: case intro.intro
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → Bool
hf : Computable f
hp : ComputablePred fun a => f a = true
⊢ Partrec fun a => Part.assert (f a = true) fun x => Part.some ()
tactic: dsimp only []
state_before: case intro.intro
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → Bool
hf : Computable f
hp : ComputablePred fun a => f a = true
⊢ Partrec fun a => Part.assert (f a = true) fun x => Part.some ()
state_after: case intro.intro
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → Bool
hf : Computable f
hp : ComputablePred fun a => f a = true
n : α
a : Unit
⊢ (a ∈ bif f n then Part.some () else Part.none) ↔ a ∈ Part.assert (f n = true) fun x => Part.some ()
tactic: refine
    (Partrec.cond hf (Decidable.Partrec.const' (Part.some ())) Partrec.none).of_eq fun n =>
      Part.ext fun a => ?_
state_before: case intro.intro
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → Bool
hf : Computable f
hp : ComputablePred fun a => f a = true
n : α
a : Unit
⊢ (a ∈ bif f n then Part.some () else Part.none) ↔ a ∈ Part.assert (f n = true) fun x => Part.some ()
state_after: case intro.intro.unit
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → Bool
hf : Computable f
hp : ComputablePred fun a => f a = true
n : α
⊢ (PUnit.unit ∈ bif f n then Part.some () else Part.none) ↔ PUnit.unit ∈ Part.assert (f n = true) fun x => Part.some ()
tactic: cases a
state_before: case intro.intro.unit
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → Bool
hf : Computable f
hp : ComputablePred fun a => f a = true
n : α
⊢ (PUnit.unit ∈ bif f n then Part.some () else Part.none) ↔ PUnit.unit ∈ Part.assert (f n = true) fun x => Part.some ()
state_after: no goals
tactic: cases f n <;> simp
Theorem: nilpotent_iff_finite_descending_central_series
File: Mathlib/GroupTheory/Nilpotent.lean

Proof steps:
state_before: G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : H.Normal
⊢ Group.IsNilpotent G ↔ ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥
state_after: G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : H.Normal
⊢ (∃ n H, IsAscendingCentralSeries H ∧ H n = ⊤) ↔ ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥
tactic: rw [nilpotent_iff_finite_ascending_central_series]
state_before: G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : H.Normal
⊢ (∃ n H, IsAscendingCentralSeries H ∧ H n = ⊤) ↔ ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥
state_after: case mp
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : H.Normal
⊢ (∃ n H, IsAscendingCentralSeries H ∧ H n = ⊤) → ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥

case mpr
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : H.Normal
⊢ (∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥) → ∃ n H, IsAscendingCentralSeries H ∧ H n = ⊤
tactic: constructor
Theorem: Filter.coprodᵢ_eq_bot_iff'
File: Mathlib/Order/Filter/Pi.lean

Proof steps:
state_before: ι : Type u_1
α : ι → Type u_2
f f₁ f₂ : (i : ι) → Filter (α i)
s : (i : ι) → Set (α i)
p : (i : ι) → α i → Prop
⊢ Filter.coprodᵢ f = ⊥ ↔ (∃ i, IsEmpty (α i)) ∨ f = ⊥
state_after: no goals
tactic: simpa only [not_neBot, not_and_or, funext_iff, not_forall, not_exists, not_nonempty_iff]
    using coprodᵢ_neBot_iff'.not
Theorem: Filter.Tendsto.div
File: Mathlib/Topology/Algebra/GroupWithZero.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : TopologicalSpace G₀
inst✝¹ : HasContinuousInv₀ G₀
inst✝ : ContinuousMul G₀
f g : α → G₀
l : Filter α
a b : G₀
hf : Tendsto f l (𝓝 a)
hg : Tendsto g l (𝓝 b)
hy : b ≠ 0
⊢ Tendsto (f / g) l (𝓝 (a / b))
state_after: no goals
tactic: simpa only [div_eq_mul_inv] using hf.mul (hg.inv₀ hy)
Theorem: Complex.exp_log
File: Mathlib/Analysis/SpecialFunctions/Complex/Log.lean

Proof steps:
state_before: x : ℂ
hx : x ≠ 0
⊢ cexp (log x) = x
state_after: no goals
tactic: rw [log, exp_add_mul_I, ← ofReal_sin, sin_arg, ← ofReal_cos, cos_arg hx, ← ofReal_exp,
    Real.exp_log (abs.pos hx), mul_add, ofReal_div, ofReal_div,
    mul_div_cancel₀ _ (ofReal_ne_zero.2 <| abs.ne_zero hx), ← mul_assoc,
    mul_div_cancel₀ _ (ofReal_ne_zero.2 <| abs.ne_zero hx), re_add_im]
Theorem: Fin.insertNth_le_iff
File: Mathlib/Data/Fin/Tuple/Basic.lean

Proof steps:
state_before: m n : ℕ
α : Fin (n + 1) → Type u
β : Type v
inst✝ : (i : Fin (n + 1)) → Preorder (α i)
i : Fin (n + 1)
x : α i
p : (j : Fin n) → α (i.succAbove j)
q : (j : Fin (n + 1)) → α j
⊢ i.insertNth x p ≤ q ↔ x ≤ q i ∧ p ≤ fun j => q (i.succAbove j)
state_after: no goals
tactic: simp [Pi.le_def, forall_iff_succAbove i]
Theorem: Multiset.aestronglyMeasurable_prod
File: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁵ : Countable ι
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f g : α → β
M : Type u_5
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
s : Multiset (α → M)
hs : ∀ f ∈ s, AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable (fun x => (Multiset.map (fun f => f x) s).prod) μ
state_after: no goals
tactic: simpa only [← Pi.multiset_prod_apply] using s.aestronglyMeasurable_prod' hs
Theorem: Real.arctan_add_arctan_lt_pi_div_two
File: Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean

Proof steps:
state_before: x y : ℝ
h : x * y < 1
⊢ arctan x + arctan y < π / 2
state_after: case inl
x y : ℝ
h : x * y < 1
hy : y ≤ 0
⊢ arctan x + arctan y < π / 2

case inr
x y : ℝ
h : x * y < 1
hy : 0 < y
⊢ arctan x + arctan y < π / 2
tactic: cases' le_or_lt y 0 with hy hy
Theorem: MonoidAlgebra.liftNC_one
File: Mathlib/Algebra/MonoidAlgebra/Basic.lean

Proof steps:
state_before: k : Type u₁
G : Type u₂
H : Type u_1
R : Type u_2
inst✝⁴ : NonAssocSemiring R
inst✝³ : Semiring k
inst✝² : One G
g_hom : Type u_3
inst✝¹ : FunLike g_hom G R
inst✝ : OneHomClass g_hom G R
f : k →+* R
g : g_hom
⊢ (liftNC ↑f ⇑g) 1 = 1
state_after: no goals
tactic: simp [one_def]
Theorem: Dilation.ratio_unique
File: Mathlib/Topology/MetricSpace/Dilation.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
F : Type u_4
G : Type u_5
inst✝³ : PseudoEMetricSpace α
inst✝² : PseudoEMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
x y : α
r : ℝ≥0
h₀ : edist x y ≠ 0
htop : edist x y ≠ ⊤
hr : edist (f x) (f y) = ↑r * edist x y
⊢ r = ratio f
state_after: no goals
tactic: simpa only [hr, ENNReal.mul_eq_mul_right h₀ htop, ENNReal.coe_inj] using edist_eq f x y
Theorem: ArithmeticFunction.abs_moebius_le_one
File: Mathlib/NumberTheory/ArithmeticFunction.lean

Proof steps:
state_before: R : Type u_1
n : ℕ
⊢ if Squarefree n then 1 ≤ 1 else 0 ≤ 1
state_after: no goals
tactic: simp
Theorem: HasDerivWithinAt.smul_const
File: Mathlib/Analysis/Calculus/Deriv/Mul.lean

Proof steps:
state_before: 𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
E : Type w
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
G : Type u_1
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s t : Set 𝕜
L L₁ L₂ : Filter 𝕜
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasDerivWithinAt c c' s x
f : F
⊢ HasDerivWithinAt (fun y => c y • f) (c' • f) s x
state_after: 𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
E : Type w
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
G : Type u_1
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s t : Set 𝕜
L L₁ L₂ : Filter 𝕜
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasDerivWithinAt c c' s x
f : F
this : HasDerivWithinAt (fun y => c y • f) (c x • 0 + c' • f) s x
⊢ HasDerivWithinAt (fun y => c y • f) (c' • f) s x
tactic: have := hc.smul (hasDerivWithinAt_const x s f)
state_before: 𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
E : Type w
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
G : Type u_1
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s t : Set 𝕜
L L₁ L₂ : Filter 𝕜
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasDerivWithinAt c c' s x
f : F
this : HasDerivWithinAt (fun y => c y • f) (c x • 0 + c' • f) s x
⊢ HasDerivWithinAt (fun y => c y • f) (c' • f) s x
state_after: no goals
tactic: rwa [smul_zero, zero_add] at this
Theorem: ComputablePred.computable_iff
File: Mathlib/Computability/Halting.lean

Proof steps:
state_before: α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
p : α → Prop
⊢ (∃ f, Computable f ∧ p = fun a => f a = true) → ComputablePred p
state_after: case intro.intro
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → Bool
h : Computable f
⊢ ComputablePred fun a => f a = true
tactic: rintro ⟨f, h, rfl⟩
state_before: case intro.intro
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → Bool
h : Computable f
⊢ ComputablePred fun a => f a = true
state_after: no goals
tactic: exact ⟨by infer_instance, by simpa using h⟩
Theorem: Option.pmap_map
File: Mathlib/Data/Option/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
p : α → Prop
f : (a : α) → p a → β
x✝ : Option α
g : γ → α
x : Option γ
H : ∀ (a : α), a ∈ Option.map g x → p a
⊢ pmap f (Option.map g x) H = pmap (fun a h => f (g a) h) x ⋯
state_after: no goals
tactic: cases x <;> simp only [map_none', map_some', pmap]
Theorem: neg_sq
File: Mathlib/Algebra/Ring/Commute.lean

Proof steps:
state_before: α : Type u
β : Type v
γ : Type w
R : Type x
inst✝¹ : Monoid R
inst✝ : HasDistribNeg R
a : R
⊢ (-a) ^ 2 = a ^ 2
state_after: no goals
tactic: simp [sq]
Theorem: Polynomial.Monic.mul_left_ne_zero
File: Mathlib/Algebra/Polynomial/Monic.lean

Proof steps:
state_before: R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p : R[X]
hp : p.Monic
q : R[X]
hq : q ≠ 0
⊢ q * p ≠ 0
state_after: case pos
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p : R[X]
hp : p.Monic
q : R[X]
hq : q ≠ 0
h : p = 1
⊢ q * p ≠ 0

case neg
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p : R[X]
hp : p.Monic
q : R[X]
hq : q ≠ 0
h : ¬p = 1
⊢ q * p ≠ 0
tactic: by_cases h : p = 1
state_before: case neg
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p : R[X]
hp : p.Monic
q : R[X]
hq : q ≠ 0
h : ¬p = 1
⊢ q * p ≠ 0
state_after: case neg
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p : R[X]
hp : p.Monic
q : R[X]
hq : q ≠ 0
h : ¬p = 1
⊢ ¬q = 0 ∧ ¬p.degree = ⊥
tactic: rw [Ne, ← degree_eq_bot, hp.degree_mul, WithBot.add_eq_bot, not_or, degree_eq_bot]
state_before: case neg
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p : R[X]
hp : p.Monic
q : R[X]
hq : q ≠ 0
h : ¬p = 1
⊢ ¬q = 0 ∧ ¬p.degree = ⊥
state_after: case neg
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p : R[X]
hp : p.Monic
q : R[X]
hq : q ≠ 0
h : ¬p = 1
⊢ ¬p.degree = ⊥
tactic: refine ⟨hq, ?_⟩
state_before: case neg
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p : R[X]
hp : p.Monic
q : R[X]
hq : q ≠ 0
h : ¬p = 1
⊢ ¬p.degree = ⊥
state_after: case neg
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p : R[X]
hp : p.Monic
q : R[X]
hq : q ≠ 0
h : 0 < p.degree
⊢ ¬p.degree = ⊥
tactic: rw [← hp.degree_le_zero_iff_eq_one, not_le] at h
state_before: case neg
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p : R[X]
hp : p.Monic
q : R[X]
hq : q ≠ 0
h : 0 < p.degree
⊢ ¬p.degree = ⊥
state_after: case neg
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p : R[X]
hp : p.Monic
q : R[X]
hq : q ≠ 0
h : 0 < p.degree
⊢ ⊥ < 0
tactic: refine (lt_trans ?_ h).ne'
state_before: case neg
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p : R[X]
hp : p.Monic
q : R[X]
hq : q ≠ 0
h : 0 < p.degree
⊢ ⊥ < 0
state_after: no goals
tactic: simp