mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case right
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z r : R
u : ↥(posSubmonoid R)
s : R
v : ↥(posSubmonoid R)
t : R
w : ↥(posSubmonoid R)
h1 : r * ↑v ≤ᵥ s * ↑u
h2 : s * ↑u ≤ᵥ r * ↑v
h3 : s * ↑w ≤ᵥ t * ↑v
h4 : t * ↑v ≤ᵥ s * ↑w
this✝ : t * ↑v * ↑u ≤ᵥ s * ↑u * ↑w
this : t * ↑u * ↑v ≤ᵥ r * ↑v * ↑w
⊢ t * ↑u ≤ᵥ r * ↑w

Tactic:
veq_mul_right_comm r

Goal state after:
case right
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z r : R
u : ↥(posSubmonoid R)
s : R
v : ↥(posSubmonoid R)
t : R
w : ↥(posSubmonoid R)
h1 : r * ↑v ≤ᵥ s * ↑u
h2 : s * ↑u ≤ᵥ r * ↑v
h3 : s * ↑w ≤ᵥ t * ↑v
h4 : t * ↑v ≤ᵥ s * ↑w
this✝ : t * ↑v * ↑u ≤ᵥ s * ↑u * ↑w
this : t * ↑u * ↑v ≤ᵥ r * ↑w * ↑v
⊢ t * ↑u ≤ᵥ r * ↑w
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case mk.mk.h₁
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z x✝¹ : R
y✝¹ : ↥(posSubmonoid R)
x✝ : R
y✝ : ↥(posSubmonoid R)
⊢ x✝¹ * x✝ * ↑(y✝ * y✝¹) ≤ᵥ x✝ * x✝¹ * ↑(y✝¹ * y✝)

Tactic:
· simp only [Submonoid.coe_mul]
      nth_grw 2 [veq_mul_comm]
      nth_grw 6 [veq_mul_comm]

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case h₂
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z x✝¹ : R
y✝¹ : ↥(posSubmonoid R)
x✝ : R
y✝ : ↥(posSubmonoid R)
⊢ x✝ * x✝¹ * ↑(y✝¹ * y✝) ≤ᵥ x✝¹ * x✝ * ↑(y✝ * y✝¹)

Tactic:
· simp only [Submonoid.coe_mul]
      nth_grw 2 [veq_mul_comm]
      nth_grw 6 [veq_mul_comm]

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case pos.refine_1
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
hw : w ≤ᵥ 0
hz : z ≤ᵥ 0
h : x * ↑v ≤ᵥ z * ↑s
⊢ z * ↑s * ↑t * ↑u ≤ᵥ 0 * (↑s * ↑v)

Tactic:
apply vle_trans (mul_vle_mul_left (mul_vle_mul_left (mul_vle_mul_left hz s) t) u)

Goal state after:
case pos.refine_1
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
hw : w ≤ᵥ 0
hz : z ≤ᵥ 0
h : x * ↑v ≤ᵥ z * ↑s
⊢ 0 * ↑s * ↑t * ↑u ≤ᵥ 0 * (↑s * ↑v)
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case neg.refine_2
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
hw : ¬w ≤ᵥ 0
hz : ¬z ≤ᵥ 0
h : y * ↑u ≤ᵥ w * ↑t
⊢ x * ↑v ≤ᵥ z * ↑s

Tactic:
apply vle_mul_cancel t.prop

Goal state after:
case neg.refine_2
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
hw : ¬w ≤ᵥ 0
hz : ¬z ≤ᵥ 0
h : y * ↑u ≤ᵥ w * ↑t
⊢ x * ↑v * ↑t ≤ᵥ z * ↑s * ↑t
Module: Mathlib.Topology.Algebra.WithZeroTopology

Goal state before:
pp: case neg
α : Type u_1
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
γ γ₁ γ₂ : Γ₀
l : Filter α
f : α → Γ₀
s t : Set Γ₀
h₁ : Disjoint (closure[topologicalSpace] s) t
h₂ : Disjoint s (closure[topologicalSpace] t)
hs : 0 ∉ s
⊢ Disjoint (𝓝ˢ s) (𝓝ˢ t)

Tactic:
· rwa [(isOpen_iff.2 (.inl hs)).nhdsSet_eq, disjoint_principal_nhdsSet]

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z x y : R
t s : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
hx : ¬x ≤ᵥ 0
hy : y ≤ᵥ 0
⊢ x ≤ᵥ 0

Tactic:
apply vle_mul_cancel t.prop

Goal state after:
case pos
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z x y : R
t s : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
hx : ¬x ≤ᵥ 0
hy : y ≤ᵥ 0
⊢ x * ↑t ≤ᵥ 0 * ↑t
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
⊢ (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | v.restrict x < ↑γ}

Tactic:
simp [Filter.hasBasis_iff, v.is_topological_valuation]

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: case refine_3
R : Type u_1
inst✝⁷ : Ring R
inst✝⁶ : ValuativeRel R
K : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : ValuativeRel K
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
cts_add : ContinuousConstVAdd R R
basis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | (valuation R).restrict x < ↑γ}
x₀ : R
⊢ ((fun x ↦ x₀ + x) ∘ fun x ↦ -x₀ + x) = _root_.id

pp: case refine_4
R : Type u_1
inst✝⁷ : Ring R
inst✝⁶ : ValuativeRel R
K : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : ValuativeRel K
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
cts_add : ContinuousConstVAdd R R
basis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | (valuation R).restrict x < ↑γ}
x₀ : R
⊢ Tendsto (fun x ↦ x₀ + x) (𝓝 0) (𝓝 x₀)

pp: case refine_5
R : Type u_1
inst✝⁷ : Ring R
inst✝⁶ : ValuativeRel R
K : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : ValuativeRel K
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
cts_add : ContinuousConstVAdd R R
basis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | (valuation R).restrict x < ↑γ}
x₀ : R
⊢ Tendsto (fun x ↦ -x₀ + x) (𝓝 x₀) (𝓝 0)

Tactic:
· ext; simp

Goal state after:
case refine_4
R : Type u_1
inst✝⁷ : Ring R
inst✝⁶ : ValuativeRel R
K : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : ValuativeRel K
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
cts_add : ContinuousConstVAdd R R
basis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | (valuation R).restrict x < ↑γ}
x₀ : R
⊢ Tendsto (fun x ↦ x₀ + x) (𝓝 0) (𝓝 x₀)

case refine_5
R : Type u_1
inst✝⁷ : Ring R
inst✝⁶ : ValuativeRel R
K : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : ValuativeRel K
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
cts_add : ContinuousConstVAdd R R
basis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | (valuation R).restrict x < ↑γ}
x₀ : R
⊢ Tendsto (fun x ↦ -x₀ + x) (𝓝 x₀) (𝓝 0)
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: case refine_1
α : Type u_3
β : Type u_4
inst✝⁵ : UniformSpace α
inst✝⁴ : AddGroup α
inst✝³ : IsUniformAddGroup α
inst✝² : UniformSpace β
inst✝¹ : AddGroup β
inst✝ : IsUniformAddGroup β
x : Completion α
⊢ IsClosed {a | (completion 0 ⋯) a = 0 a}

Tactic:
apply isClosed_eq (AddMonoidHom.continuous_completion (0 : α →+ β) continuous_const)
    exact continuous_const

Goal state after:
Module: Mathlib.Topology.Algebra.UniformRing

Goal state before:
pp: α : Type u_1
inst✝⁷ : Ring α
inst✝⁶ : UniformSpace α
inst✝⁵ : IsTopologicalRing α
inst✝⁴ : IsUniformAddGroup α
β : Type u
inst✝³ : UniformSpace β
inst✝² : Ring β
inst✝¹ : IsUniformAddGroup β
inst✝ : IsTopologicalRing β
f✝ : α →+* β
hf✝ : Continuous ⇑f✝
f : α ≃+* β
hf : Continuous ⇑f
hf' : Continuous ⇑f.symm
⊢ (mapRingHom f.toRingHom hf).comp (mapRingHom f.symm.toRingHom hf') = RingHom.id (Completion β)

Tactic:
simp [mapRingHom_comp]

Goal state after:
Module: Mathlib.Topology.Algebra.UniformRing

Goal state before:
pp: α : Type u_1
inst✝⁷ : Ring α
inst✝⁶ : UniformSpace α
inst✝⁵ : IsTopologicalRing α
inst✝⁴ : IsUniformAddGroup α
β : Type u
inst✝³ : UniformSpace β
inst✝² : Ring β
inst✝¹ : IsUniformAddGroup β
inst✝ : IsTopologicalRing β
f✝ : α →+* β
hf✝ : Continuous ⇑f✝
f : α ≃+* β
hf : Continuous ⇑f
hf' : Continuous ⇑f.symm
⊢ (mapRingHom f.symm.toRingHom hf').comp (mapRingHom f.toRingHom hf) = RingHom.id (Completion α)

Tactic:
simp [mapRingHom_comp]

Goal state after:
Module: Mathlib.Topology.Algebra.UniformRing

Goal state before:
pp: α : Type u_1
inst✝¹¹ : Ring α
inst✝¹⁰ : UniformSpace α
inst✝⁹ : IsTopologicalRing α
inst✝⁸ : IsUniformAddGroup α
β : Type u
inst✝⁷ : UniformSpace β
inst✝⁶ : Ring β
inst✝⁵ : IsUniformAddGroup β
inst✝⁴ : IsTopologicalRing β
f : α →+* β
hf : Continuous ⇑f
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
inst✝¹ : IsUniformAddGroup R
inst✝ : IsTopologicalRing R
⊢ Algebra R (Completion R)

Tactic:
by infer_instance

Goal state after:
Module: Mathlib.RingTheory.Valuation.Discrete.Basic

Goal state before:
pp: case h
Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
r : ↥v.valuationSubring
hr : r ≠ 0
π : v.Uniformizer
hr₀ : v ↑r ≠ 0
vr : Γˣ := Units.mk0 (v ↑r) hr₀
hvr_def : vr = Units.mk0 (v ↑r) hr₀
m : ℤ
hm : Units.mk0 (v ↑π.val) ⋯ ^ m = vr
hm' : v ↑π.val ^ m = v ↑r
hm₀ : 0 ≤ m
n : ℕ
hn : m = ↑n
hpow : v (↑π.val ^ (-m) * ↑r) = 1
a : ↥v.valuationSubring := ⟨↑π.val ^ (-m) * ↑r, ⋯⟩
ha : a = ⟨↑π.val ^ (-m) * ↑r, ⋯⟩
ha₀ : ↑a ≠ 0
⊢ ∃ u, ↑r = ↑(π.val ^ n) * ↑↑u

Tactic:
have h_unit_a : IsUnit a :=
    Integers.isUnit_of_one (integer.integers v) (isUnit_iff_ne_zero.mpr ha₀) hpow

Goal state after:
case h
Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
r : ↥v.valuationSubring
hr : r ≠ 0
π : v.Uniformizer
hr₀ : v ↑r ≠ 0
vr : Γˣ := Units.mk0 (v ↑r) hr₀
hvr_def : vr = Units.mk0 (v ↑r) hr₀
m : ℤ
hm : Units.mk0 (v ↑π.val) ⋯ ^ m = vr
hm' : v ↑π.val ^ m = v ↑r
hm₀ : 0 ≤ m
n : ℕ
hn : m = ↑n
hpow : v (↑π.val ^ (-m) * ↑r) = 1
a : ↥v.valuationSubring := ⟨↑π.val ^ (-m) * ↑r, ⋯⟩
ha : a = ⟨↑π.val ^ (-m) * ↑r, ⋯⟩
ha₀ : ↑a ≠ 0
h_unit_a : IsUnit a
⊢ ∃ u, ↑r = ↑(π.val ^ n) * ↑↑u
Module: Mathlib.RingTheory.Valuation.Discrete.Basic

Goal state before:
pp: Γ : Type u_1
inst✝³ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝² : Field K
v : Valuation K Γ
inst✝¹ : IsCyclic ↥(MonoidWithZeroHom.ofClass v).valueGroup
inst✝ : Nontrivial ↥(MonoidWithZeroHom.ofClass v).valueGroup
I P : Ideal ↥v.valuationSubring
hP : P.IsPrime
h_ne_bot : ¬P = ⊥
⊢ Nonempty v.Uniformizer

Tactic:
by infer_instance

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
⊢ IsTopologicalRing K

Tactic:
by infer_instance

Goal state after:
Module: Mathlib.Order.Interval.Finset.DenselyOrdered

Goal state before:
pp: X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
H : DenselyOrdered X
⊢ Subsingleton X

Tactic:
← not_nontrivial_iff_subsingleton,

Goal state after:
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
H : DenselyOrdered X
⊢ ¬Nontrivial X
Module: Mathlib.Order.Interval.Finset.DenselyOrdered

Goal state before:
pp: X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H : DenselyOrdered ↑s
⊢ Subsingleton ↑s

Tactic:
← not_nontrivial_iff_subsingleton,

Goal state after:
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H : DenselyOrdered ↑s
⊢ ¬Nontrivial ↑s
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
this : T2Space K
⊢ T0Space K

Tactic:
by infer_instance

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
γ : Γ₀
hγ : γ ≠ 0
x : K
hx : v x = γ
⊢ (restrict₀ (ofClass v)) x ≠ 0

Tactic:
by simp [restrict₀_apply, hx, hγ]

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
f : G ≃+o ℤ
this : LocallyFiniteOrder G
⊢ 0 ∈ lowerBounds {x | 0 ≤ x}

Tactic:
by simp [mem_lowerBounds]

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case pos
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
hab : extensionValuation.restrict ↑⋯.choose = extensionValuation.restrict ↑⋯.choose
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
hx : v x = embedding a
hy : v y = embedding b
this : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
ha0 : a = 0
h0 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ embedding a = embedding b

Tactic:
simp [h0, reduceDIte, extension_eq_zero_iff, map_eq_zero,
        left_eq_dite_iff, WithZero.zero_ne_coe, imp_false, not_not] at this

Goal state after:
case pos
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
hab : extensionValuation.restrict ↑⋯.choose = extensionValuation.restrict ↑⋯.choose
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
hx : v x = embedding a
hy : v y = embedding b
ha0 : a = 0
h0 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : y = 0
⊢ embedding a = embedding b
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
⊢ v.intValuation r < 1 ↔ v.asIdeal ∣ Ideal.span {r}

Tactic:
by
  by_cases hr : r = 0
  · simp [hr]
  · rw [v.intValuation_if_neg hr, ← exp_zero, exp_lt_exp,
      neg_lt_zero, ← Int.ofNat_zero, Int.ofNat_lt, zero_lt_iff]
    have h : (Ideal.span {r} : Ideal R) ≠ 0 := by
      rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
      exact hr
    exact Associates.count_ne_zero_iff_dvd h v.irreducible

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
w : σ → ℕ
d : σ →₀ ℕ
a : R
inst✝ : Decidable (a = 0)
h : a = 0
⊢ weightedOrder w ((monomial d) 0) = ⊤

Tactic:
weightedOrder_eq_top_iff,

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
w : σ → ℕ
d : σ →₀ ℕ
a : R
inst✝ : Decidable (a = 0)
h : a = 0
⊢ (monomial d) 0 = 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x : K
hv : (valuation K v) x ≤ 1
γ : (WithZero (Multiplicative ℤ))ˣ
n d : R
hd✝ : d ∈ v.asIdeal.primeCompl
hd : v.intValuation d = 1
hnd : x * (algebraMap R K) ↑⟨d, hd✝⟩ = (algebraMap R K) n
hd' : v.intValuation n ≤ v.intValuation d
a : R
hval : v.intValuation (n - a * d) < ↑γ
h : (valuation K v) ((algebraMap R K) a - x) = v.intValuation (n - a * d)
⊢ (valuation K v) ((algebraMap R K) a - x) < ↑γ

Tactic:
by rwa [h]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
w : σ → ℕ
R : Type u_3
inst✝ : Ring R
p : MvPowerSeries σ R
T : Subring R
hp : ∀ (n : σ →₀ ℕ), (coeff n) p ∈ T
d : σ →₀ ℕ
hd : ↑((weight w) d) < weightedOrder w p
⊢ (coeff d) (p.toSubring T hp) = 0

Tactic:
exact_mod_cast (coeff_toSubring p T hp) ▸ (coeff_eq_zero_of_lt_weightedOrder w hd)

Goal state after:
Module: Mathlib.Topology.Algebra.LinearTopology

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁷ : Ring R
inst✝⁶ : Ring R'
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module R' M
inst✝² : SMulCommClass R R' M
inst✝¹ : TopologicalSpace M
inst✝ : DiscreteTopology M
⊢ (𝓝 0).HasBasis (fun x ↦ True) fun x ↦ ↑⊥

Tactic:
rw [nhds_discrete]

Goal state after:
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁷ : Ring R
inst✝⁶ : Ring R'
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module R' M
inst✝² : SMulCommClass R R' M
inst✝¹ : TopologicalSpace M
inst✝ : DiscreteTopology M
⊢ (pure 0).HasBasis (fun x ↦ True) fun x ↦ ↑⊥
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝³ : Field K
inst✝² : CommSemiring S
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x : R
y : ↥(adicCompletionIntegers K v)
z : adicCompletion K v
⊢ (x • y) • z = x • y • z

Tactic:
simp only [Algebra.smul_def]; apply mul_assoc

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Trunc

Goal state before:
pp: case succ
R : Type u_2
inst✝ : CommSemiring R
f : R⟦X⟧
n a : ℕ
ih : (trunc n) (↑((trunc n) f) ^ a) = (trunc n) (f ^ a)
⊢ (trunc n) (↑((trunc n) f) ^ (a + 1)) = (trunc n) (f ^ (a + 1))

Tactic:
_root_.pow_succ',

Goal state after:
case succ
R : Type u_2
inst✝ : CommSemiring R
f : R⟦X⟧
n a : ℕ
ih : (trunc n) (↑((trunc n) f) ^ a) = (trunc n) (f ^ a)
⊢ (trunc n) (↑((trunc n) f) * ↑((trunc n) f) ^ a) = (trunc n) (f ^ (a + 1))
Module: Mathlib.RingTheory.PowerSeries.Trunc

Goal state before:
pp: case succ
R : Type u_2
inst✝ : CommSemiring R
f : R⟦X⟧
n a : ℕ
ih : (trunc n) (↑((trunc n) f) ^ a) = (trunc n) (f ^ a)
⊢ (trunc n) (↑((trunc n) f) * ↑((trunc n) f) ^ a) = (trunc n) (f ^ (a + 1))

Tactic:
_root_.pow_succ',

Goal state after:
case succ
R : Type u_2
inst✝ : CommSemiring R
f : R⟦X⟧
n a : ℕ
ih : (trunc n) (↑((trunc n) f) ^ a) = (trunc n) (f ^ a)
⊢ (trunc n) (↑((trunc n) f) * ↑((trunc n) f) ^ a) = (trunc n) (f * f ^ a)
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: case e_l₂
R : Type u_1
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
u : Filter ι
g : R⟦X⟧
d : Unit →₀ ℕ
⊢ d = Finsupp.single () (d ())

Tactic:
· ext; simp

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: case e'_5.e_5
R : Type u_1
inst✝¹ : Semiring R
inst✝ : TopologicalSpace R
f : R⟦X⟧
x✝ : Unit →₀ ℕ
⊢ Finsupp.single () ((Finsupp.uniqueLinearEquiv ℕ ℕ ()) x✝) = x✝

pp: case e'_5.e_6
R : Type u_1
inst✝¹ : Semiring R
inst✝ : TopologicalSpace R
f : R⟦X⟧
x✝ : Unit →₀ ℕ
⊢ Finsupp.single () ((Finsupp.uniqueLinearEquiv ℕ ℕ ()) x✝) = x✝

Tactic:
all_goals { ext; simp }

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: case mp.right
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
h' : ¬e ≤ d
x : R
a✝ : x ∈ ⊤
⊢ x ∈ K

Tactic:
have (d') (hd'_le : d' ≤ d) : coeff d' (monomial e x) ∈ J := by
        rw [coeff_monomial]
        split_ifs with hd' <;> [exact (h' (hd' ▸ hd'_le)).elim; exact J.zero_mem]

Goal state after:
case mp.right
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
h' : ¬e ≤ d
x : R
a✝ : x ∈ ⊤
this : ∀ d' ≤ d, (coeff d') ((monomial e) x) ∈ J
⊢ x ∈ K
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case true
c : ℝ
f g : ℕ → Bool
h1 : 0 ≤ c
h2 : c < 1
n : ℕ
h : f n = true
h3 : g n = true
⊢ cantorFunctionAux c f n ≤ cantorFunctionAux c g n

Tactic:
simp [h, h3]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: ⊢ #↑Set.univ = 𝔠

Tactic:
rw [mk_univ, mk_real]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case e'_3
a : ℝ
h : #↑(Ioi a) < 𝔠
⊢ Iio a ∪ {a} = Iic a

Tactic:
exact Iio_union_right

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ (algebraMap R S) ((coeff e) f) = (coeff e) f • 1

Tactic:
algebra_compatible_smul S,

Goal state after:
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ (algebraMap R S) ((coeff e) f) = (algebraMap R S) ((coeff e) f) • 1
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
f : MvPowerSeries σ R
hf : IsNilpotent (constantCoeff f)
⊢ IsNilpotent (constantCoeff ((substAlgHom ha) f))

Tactic:
by
  simpa using IsNilpotent_subst ha hf

Goal state after:
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : Module 𝕜 E
inst✝³ : AddCommGroup F
inst✝² : TopologicalSpace F
inst✝¹ : Module 𝕜 F
f : α → E
g : α → F
inst✝ : TopologicalSpace α
x : α
s : Set α
h : f x = 0
⊢ f =o[𝕜; pure x] g

Tactic:
exact .congr' (.zero g _) h.symm .rfl

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f f₁ : 𝕜 → F
x : 𝕜
s : Set 𝕜
hs : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
⊢ derivWithin f₁ s x = derivWithin f s x

Tactic:
by
  unfold derivWithin
  rw [hs.fderivWithin_eq hx]

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : 𝕜 → F
s : Set 𝕜
hfg : EqOn f g s
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
x : 𝕜
hx : x ∈ s
⊢ derivWithin f s x = _root_.deriv g x

Tactic:
← derivWithin_of_isOpen hs hx

Goal state after:
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : 𝕜 → F
s : Set 𝕜
hfg : EqOn f g s
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
x : 𝕜
hx : x ∈ s
⊢ derivWithin f s x = derivWithin g s x