mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: K : Type u_7
inst✝¹ : DivisionRing K
Γ₀ : Type u_8
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation K Γ₀
γ : (ofClass v).ValueGroup₀ˣ
r : K
hr : (restrict₀ (ofClass v)) r = ↑γ
⊢ 0 < v r

Tactic:
← map_zero v

Goal state after:
K : Type u_7
inst✝¹ : DivisionRing K
Γ₀ : Type u_8
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation K Γ₀
γ : (ofClass v).ValueGroup₀ˣ
r : K
hr : (restrict₀ (ofClass v)) r = ↑γ
⊢ v 0 < v r
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
x : K
h : x ≠ 0
h0 : v x ≠ 0
⊢ v x ≠ 0

Tactic:
by simpa using h0

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_2
G : Type u_1
G' : Type u_2
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : CommGroup G'
inst✝¹ : LinearOrder G'
inst✝ : IsOrderedMonoid G'
x : G
y : G'
hxy : x = 1 ↔ y = 1
hx : ¬x = 1
x' : G := max x x⁻¹
hx' : x' = max x x⁻¹
xpos : 1 < x'
y' : G' := max y y⁻¹
hy' : y' = max y y⁻¹
ypos : 1 < y'
hxc : closure {x} = closure {x'}
hyc : closure {y} = closure {y'}
Hx : ∀ (z : ↥(closure {x})), z ∈ zpowers ⟨x', ⋯⟩
Hy : ∀ (z : ↥(closure {y})), z ∈ zpowers ⟨y', ⋯⟩
m n : ℤ
⊢ (mulEquivOfOrderOfEq Hx Hy ⋯).toFun ((fun x_1 ↦ ⟨x', ⋯⟩ ^ x_1) m) ≤
      (mulEquivOfOrderOfEq Hx Hy ⋯).toFun ((fun x_1 ↦ ⟨x', ⋯⟩ ^ x_1) n) ↔
    (fun x_1 ↦ ⟨x', ⋯⟩ ^ x_1) m ≤ (fun x_1 ↦ ⟨x', ⋯⟩ ^ x_1) n

Tactic:
simp only [MulEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe, map_zpow,
        mulEquivOfOrderOfEq_apply_gen]

Goal state after:
case refine_2
G : Type u_1
G' : Type u_2
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : CommGroup G'
inst✝¹ : LinearOrder G'
inst✝ : IsOrderedMonoid G'
x : G
y : G'
hxy : x = 1 ↔ y = 1
hx : ¬x = 1
x' : G := max x x⁻¹
hx' : x' = max x x⁻¹
xpos : 1 < x'
y' : G' := max y y⁻¹
hy' : y' = max y y⁻¹
ypos : 1 < y'
hxc : closure {x} = closure {x'}
hyc : closure {y} = closure {y'}
Hx : ∀ (z : ↥(closure {x})), z ∈ zpowers ⟨x', ⋯⟩
Hy : ∀ (z : ↥(closure {y})), z ∈ zpowers ⟨y', ⋯⟩
m n : ℤ
⊢ ⟨y', ⋯⟩ ^ m ≤ ⟨y', ⋯⟩ ^ n ↔ ⟨x', ⋯⟩ ^ m ≤ ⟨x', ⋯⟩ ^ n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsDiscreteValuationRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
W : WeierstrassCurve K
inst✝ : IsMinimal R W
⊢ IsMinimal R W ∧ (valuation K (IsDiscreteValuationRing.maximalIdeal R)) ((algebraMap R K) (integralModel R W).Δ) = 1 ∨
    IsMinimal R W ∧
        (valuation K (IsDiscreteValuationRing.maximalIdeal R)) ((algebraMap R K) (integralModel R W).Δ) < 1 ∧
          (valuation K (IsDiscreteValuationRing.maximalIdeal R)) ((algebraMap R K) (integralModel R W).c₄) = 1 ∨
      IsMinimal R W ∧
        (valuation K (IsDiscreteValuationRing.maximalIdeal R)) ((algebraMap R K) (integralModel R W).Δ) < 1 ∧
          (valuation K (IsDiscreteValuationRing.maximalIdeal R)) ((algebraMap R K) (integralModel R W).c₄) < 1

Tactic:
grind [valuation_le_one]

Goal state after:
Module: Mathlib.Order.Northcott

Goal state before:
pp: α : Type u_1
β : Type u_2
h : α → β
inst✝¹ : LinearOrder β
inst✝ : NoMaxOrder β
⊢ Northcott h ↔ Tendsto h cofinite atTop

Tactic:
northcott_iff,

Goal state after:
α : Type u_1
β : Type u_2
h : α → β
inst✝¹ : LinearOrder β
inst✝ : NoMaxOrder β
⊢ (∀ (b : β), {a | h a ≤ b}.Finite) ↔ Tendsto h cofinite atTop
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
n : σ →₀ ℕ
a : R
⊢ (coeff n) ((monomial n) a) = a

Tactic:
rw [monomial_def]
  exact Pi.single_eq_same _ _

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
⊢ (monomial 0) ↑0 = 0

Tactic:
simp [Nat.cast]

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

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : k = m + n
⊢ 0 = a * b

Tactic:
rw [h₂] at h₁
    exact (h₁ <| le_add_left le_rfl).elim

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

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : Nontrivial R
s t : σ
H : ¬single s 1 = single t 1
h : 1 = 0
⊢ s = t

Tactic:
· exfalso
      exact one_ne_zero h

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

Goal state before:
pp: case insert
σ : Type u_1
R : Type u_3
inst✝² : CommSemiring R
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq σ
f : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : ∀ (d : σ →₀ ℕ), (coeff d) (∏ j ∈ s, f j) = ∑ l ∈ s.finsuppAntidiag d, ∏ i ∈ s, (coeff (l i)) (f i)
d : σ →₀ ℕ
⊢ (↑(antidiagonal d)).PairwiseDisjoint fun p ↦
    Finset.map { toFun := fun f ↦ (↑f).update a p.1, inj' := ⋯ } (s.finsuppAntidiag p.2).attach

Tactic:
· simp only [Set.PairwiseDisjoint, Set.Pairwise, mem_coe, mem_antidiagonal, ne_eq,
        disjoint_left, mem_map, mem_attach, Function.Embedding.coeFn_mk, true_and, Subtype.exists,
        exists_prop, not_exists, not_and, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂,
        Prod.forall, Prod.mk.injEq]
      rintro u v rfl u' v' huv h k - l - hkl
      obtain rfl : u' = u := by
        simpa only [Finsupp.coe_update, Function.update_self] using DFunLike.congr_fun hkl a
      simp only [add_right_inj] at huv
      exact h rfl huv.symm

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case inr
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
n d : R
hd : d ∈ nonZeroDivisors R
h : ∀ (v : HeightOneSpectrum R), (valuation K v) (IsLocalization.mk' K n ⟨d, hd⟩) ≤ 1
hx : IsLocalization.mk' K n ⟨d, hd⟩ * (algebraMap R K) ↑(n, ⟨d, hd⟩).2 = (algebraMap R K) (n, ⟨d, hd⟩).1
hn0 : n ≠ 0
hd0 : d ≠ 0
ine : ∀ {r : R}, r ≠ 0 → Ideal.span {r} ≠ ⊥
v : Ideal R
hv : Irreducible (Associates.mk v)
⊢ Multiset.count ⟨Associates.mk v, hv⟩ (Associates.factors' (Ideal.span {d})) ≤
    Multiset.count ⟨Associates.mk v, hv⟩ (Associates.factors' (Ideal.span {n}))

Tactic:
have hv' := hv

Goal state after:
case inr
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
n d : R
hd : d ∈ nonZeroDivisors R
h : ∀ (v : HeightOneSpectrum R), (valuation K v) (IsLocalization.mk' K n ⟨d, hd⟩) ≤ 1
hx : IsLocalization.mk' K n ⟨d, hd⟩ * (algebraMap R K) ↑(n, ⟨d, hd⟩).2 = (algebraMap R K) (n, ⟨d, hd⟩).1
hn0 : n ≠ 0
hd0 : d ≠ 0
ine : ∀ {r : R}, r ≠ 0 → Ideal.span {r} ≠ ⊥
v : Ideal R
hv hv' : Irreducible (Associates.mk v)
⊢ Multiset.count ⟨Associates.mk v, hv⟩ (Associates.factors' (Ideal.span {d})) ≤
    Multiset.count ⟨Associates.mk v, hv⟩ (Associates.factors' (Ideal.span {n}))
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
h : Function.Surjective ⇑v
γ : Γ₀
H : ¬∃ x, v x = γ
hγ : γ ≠ 0
r : hat K
hr : v r = γ
hr' : (restrict₀ (ofClass v)) r ≠ 0
x : hat K
⊢ x ∈ {a | ¬v a = γ} ↔ x ∈ univ \ {x | v.restrict x = (restrict₀ (ofClass v)) r}

Tactic:
simp [← hr, ← v.restrict_def, v.restrict_inj]

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
H : weightedOrder w f < weightedOrder w g
n : ℕ
hn : ↑n = weightedOrder w f
d : σ →₀ ℕ
hd' : (coeff d) f ≠ 0
hd : (weight w) d = n
b : σ →₀ ℕ
hb : (weight w) b < n
this : ↑((weight w) b) < weightedOrder w f
⊢ (coeff b) f + (coeff b) g = 0

Tactic:
coeff_eq_zero_of_lt_weightedOrder w this,

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
H : weightedOrder w f < weightedOrder w g
n : ℕ
hn : ↑n = weightedOrder w f
d : σ →₀ ℕ
hd' : (coeff d) f ≠ 0
hd : (weight w) d = n
b : σ →₀ ℕ
hb : (weight w) b < n
this : ↑((weight w) b) < weightedOrder w f
⊢ 0 + (coeff b) g = 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
⊢ ∀ (d : σ →₀ ℕ), ↑((weight w) d) < weightedOrder w f + weightedOrder w g → (coeff d) (f * g) = 0

Tactic:
intro d hd

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g
⊢ (coeff d) (f * g) = 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g
i j : σ →₀ ℕ
hij : (i, j) ∈ Finset.antidiagonal d
hi : weightedOrder w f ≤ ↑((weight w) i)
⊢ (coeff (i, j).1) f * (coeff (i, j).2) g = 0

Tactic:
by_cases! hj : weight w j < g.weightedOrder w
    · rw [coeff_eq_zero_of_lt_weightedOrder w hj, mul_zero]
    · simp only [Finset.mem_antidiagonal] at hij
      exfalso
      apply ne_of_lt (lt_of_lt_of_le hd <| add_le_add hi hj)
      rw [← hij, map_add, Nat.cast_add]

Goal state after:
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
⊢ (ofClass (valuation K v)).valueGroup = (ofClass Valued.v).valueGroup

Tactic:
by
  simp [valuation, valueGroup, valueMonoid, ← (toCompletion_surjective K v).range_comp]; rfl

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: x : ℕ∞
hx : x ≠ ⊤
⊢ IsOpen {x}

Tactic:
rw [isOpen_singleton_iff_nhds_eq_pure, ENat.nhds_eq_pure hx]

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
⊢ f ≠ 0 ↔ ∃ n d, (coeff d) f ≠ 0 ∧ degree d = n

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
⊢ f ≠ 0 ↔ ∃ n d, (coeff d) f ≠ 0 ∧ (weight fun x ↦ 1) d = n
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
h : ↑f.order.toNat = f.order
⊢ ∃ d, (coeff d) f ≠ 0 ∧ ↑(degree d) = f.order

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
h : ↑f.order.toNat = f.order
⊢ ∃ d, (coeff d) f ≠ 0 ∧ ↑((weight fun x ↦ 1) d) = f.order
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
d : σ →₀ ℕ
h : (coeff d) f ≠ 0
⊢ f.order ≤ ↑(degree d)

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
d : σ →₀ ℕ
h : (coeff d) f ≠ 0
⊢ f.order ≤ ↑((weight fun x ↦ 1) d)
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑(degree d) < f.order
⊢ (coeff d) f = 0

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑((weight fun x ↦ 1) d) < f.order
⊢ (coeff d) f = 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
n : ℕ
h : ∀ (d : σ →₀ ℕ), degree d < n → (coeff d) f = 0
⊢ ↑n ≤ f.order

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
n : ℕ
h : ∀ (d : σ →₀ ℕ), (weight fun x ↦ 1) d < n → (coeff d) f = 0
⊢ ↑n ≤ f.order
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
n : ℕ∞
h : ∀ (d : σ →₀ ℕ), ↑(degree d) < n → (coeff d) f = 0
⊢ n ≤ f.order

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
n : ℕ∞
h : ∀ (d : σ →₀ ℕ), ↑((weight fun x ↦ 1) d) < n → (coeff d) f = 0
⊢ n ≤ f.order
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
n : ℕ
⊢ f.order = ↑n ↔ (∃ d, (coeff d) f ≠ 0 ∧ degree d = n) ∧ ∀ (d : σ →₀ ℕ), degree d < n → (coeff d) f = 0

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
n : ℕ
⊢ f.order = ↑n ↔
    (∃ d, (coeff d) f ≠ 0 ∧ (weight fun x ↦ 1) d = n) ∧ ∀ (d : σ →₀ ℕ), (weight fun x ↦ 1) d < n → (coeff d) f = 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
d : σ →₀ ℕ
a : R
inst✝ : Decidable (a = 0)
⊢ ((monomial d) a).order = if a = 0 then ⊤ else ↑(degree d)

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
d : σ →₀ ℕ
a : R
inst✝ : Decidable (a = 0)
⊢ ((monomial d) a).order = if a = 0 then ⊤ else ↑((weight fun x ↦ 1) d)
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
d : σ →₀ ℕ
a : R
h : a ≠ 0
⊢ ((monomial d) a).order = ↑(degree d)

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
d : σ →₀ ℕ
a : R
h : a ≠ 0
⊢ ((monomial d) a).order = ↑((weight fun x ↦ 1) d)
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: case ind.a.e'_5
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
k : ℕ
ih : ∀ m ≤ k, ∀ {x : σ →₀ ℕ}, degree x < n → MvPolynomial.coeff x ((truncTotal n) p ^ m) = (coeff x) (p ^ m)
x : σ →₀ ℕ
h : degree x < n
x✝ : (σ →₀ ℕ) × (σ →₀ ℕ)
a✝ : x✝ ∈ antidiagonal x
⊢ MvPolynomial.coeff x✝.1 ((truncTotal n) p ^ k) = (coeff x✝.1) (p ^ k)

pp: case ind.a.e'_6
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
k : ℕ
ih : ∀ m ≤ k, ∀ {x : σ →₀ ℕ}, degree x < n → MvPolynomial.coeff x ((truncTotal n) p ^ m) = (coeff x) (p ^ m)
x : σ →₀ ℕ
h : degree x < n
x✝ : (σ →₀ ℕ) × (σ →₀ ℕ)
a✝ : x✝ ∈ antidiagonal x
⊢ MvPolynomial.coeff x✝.2 ((truncTotal n) p) = (coeff x✝.2) p

Tactic:
· exact ih _ k.le_refl (by grind [mem_antidiagonal])

Goal state after:
case ind.a.e'_6
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
k : ℕ
ih : ∀ m ≤ k, ∀ {x : σ →₀ ℕ}, degree x < n → MvPolynomial.coeff x ((truncTotal n) p ^ m) = (coeff x) (p ^ m)
x : σ →₀ ℕ
h : degree x < n
x✝ : (σ →₀ ℕ) × (σ →₀ ℕ)
a✝ : x✝ ∈ antidiagonal x
⊢ MvPolynomial.coeff x✝.2 ((truncTotal n) p) = (coeff x✝.2) p
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : Ring R
f g : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑(degree d) < g.order
⊢ (coeff d) (f * (1 - g)) = (coeff d) f

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : Ring R
f g : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑((weight fun x ↦ 1) d) < g.order
⊢ (coeff d) (f * (1 - g)) = (coeff d) f
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : Ring R
f g : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑(degree d) < g.order
⊢ (coeff d) ((1 - g) * f) = (coeff d) f

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : Ring R
f g : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑((weight fun x ↦ 1) d) < g.order
⊢ (coeff d) ((1 - g) * f) = (coeff d) f
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_4
ι : Type u_5
inst✝ : CommRing R
d : σ →₀ ℕ
s : Finset ι
f : MvPowerSeries σ R
g : ι → MvPowerSeries σ R
⊢ (∀ i ∈ s, ↑(degree d) < (g i).order) → (coeff d) (f * ∏ i ∈ s, (1 - g i)) = (coeff d) f

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_4
ι : Type u_5
inst✝ : CommRing R
d : σ →₀ ℕ
s : Finset ι
f : MvPowerSeries σ R
g : ι → MvPowerSeries σ R
⊢ (∀ i ∈ s, ↑((weight fun x ↦ 1) d) < (g i).order) → (coeff d) (f * ∏ i ∈ s, (1 - g i)) = (coeff d) f
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_4
ι : Type u_5
inst✝ : CommRing R
d : σ →₀ ℕ
s : Finset ι
f : MvPowerSeries σ R
g : ι → MvPowerSeries σ R
⊢ (∀ i ∈ s, ↑((weight fun x ↦ 1) d) < (g i).order) → (coeff d) (f * ∏ i ∈ s, (1 - g i)) = (coeff d) f

Tactic:
exact coeff_mul_prod_one_sub_of_lt_weightedOrder _ d s f g

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
p : ℕ
hf : f.IsHomogeneous p
d : σ →₀ ℕ
hd : degree d ≠ p
⊢ (weight 1) d ≠ p

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
p : ℕ
hf : f.IsHomogeneous p
d : σ →₀ ℕ
hd : (weight fun x ↦ 1) d ≠ p
⊢ (weight 1) d ≠ p
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
p : ℕ
d : σ →₀ ℕ
f : MvPowerSeries σ R
⊢ (coeff d) ((homogeneousComponent p) f) = if degree d = p then (coeff d) f else 0

Tactic:
degree_eq_weight_one

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
p : ℕ
d : σ →₀ ℕ
f : MvPowerSeries σ R
⊢ (coeff d) ((homogeneousComponent p) f) = if (weight fun x ↦ 1) d = p then (coeff d) f else 0
Module: Mathlib.RingTheory.MvPowerSeries.PiTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
w : σ → ℕ
h : Tendsto (fun i ↦ weightedOrder w (f i)) atTop (𝓝 ⊤)
⊢ Summable f

Tactic:
rcases isEmpty_or_nonempty ι with hempty | hempty
  · apply summable_empty
  rw [summable_iff_summable_coeff]
  simp_rw [ENat.tendsto_nhds_top_iff_natCast_lt, Filter.eventually_atTop] at h
  intro d
  obtain ⟨i, hi⟩ := h (Finsupp.weight w d)
  refine summable_of_hasFiniteSupport <| (Set.finite_Iic i).subset ?_
  simp_rw [Function.support_subset_iff, Set.mem_Iic]
  intro k hk
  contrapose! hk
  exact coeff_eq_zero_of_lt_weightedOrder w <| hi k hk.le

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

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
f : R⟦X⟧
n : ℕ
⊢ (coeff n) (X ^ f.order.toNat * f.divXPowOrder) = (coeff n) f

Tactic:
rw [coeff_X_pow_mul']

Goal state after:
R : Type u_1
inst✝ : Semiring R
f : R⟦X⟧
n : ℕ
⊢ (if f.order.toNat ≤ n then (coeff (n - f.order.toNat)) f.divXPowOrder else 0) = (coeff n) f
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_2
inst✝ : Semiring R
φ : R⟦X⟧
hφ : φ ≠ 0
n : ℕ
ho : φ.order = ↑n
hn : φ.order.toNat = n
ψ : R⟦X⟧
H : φ = X ^ (φ.order.toNat + 1) * ψ
this : (coeff n) φ = 0
⊢ False

pp: R : Type u_2
inst✝ : Semiring R
φ : R⟦X⟧
hφ : φ ≠ 0
n : ℕ
ho : φ.order = ↑n
hn : φ.order.toNat = n
ψ : R⟦X⟧
H : φ = X ^ (φ.order.toNat + 1) * ψ
this : (coeff n) φ = (coeff n) (ψ * X ^ (φ.order.toNat + 1))
⊢ ↑n < (X ^ (φ.order.toNat + 1)).order

Tactic:
← hn

Goal state after:
R : Type u_2
inst✝ : Semiring R
φ : R⟦X⟧
hφ : φ ≠ 0
n : ℕ
ho : φ.order = ↑n
hn : φ.order.toNat = n
ψ : R⟦X⟧
H : φ = X ^ (φ.order.toNat + 1) * ψ
this : (coeff φ.order.toNat) φ = 0
⊢ False

R : Type u_2
inst✝ : Semiring R
φ : R⟦X⟧
hφ : φ ≠ 0
n : ℕ
ho : φ.order = ↑n
hn : φ.order.toNat = n
ψ : R⟦X⟧
H : φ = X ^ (φ.order.toNat + 1) * ψ
this : (coeff n) φ = (coeff n) (ψ * X ^ (φ.order.toNat + 1))
⊢ ↑n < (X ^ (φ.order.toNat + 1)).order
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : UniformSpace R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : IsUniformAddGroup R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : CompleteSpace S
inst✝² : T2Space S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁹.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑φ
ha : HasEval a
f : MvPowerSeries σ R
h : ∃ p, ↑p = f
⊢ ⋯.extend (MvPolynomial.eval₂ φ a) f = MvPolynomial.eval₂ φ a h.choose

Tactic:
obtain ⟨p, rfl⟩ := h

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : UniformSpace R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : IsUniformAddGroup R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : CompleteSpace S
inst✝² : T2Space S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁹.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑φ
ha : HasEval a
p : MvPolynomial σ R
⊢ ⋯.extend (MvPolynomial.eval₂ φ a) ↑p = MvPolynomial.eval₂ φ a ⋯.choose
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
φ : R →+* S
a : S
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformSpace S
inst✝⁶ : IsUniformAddGroup R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : T2Space S
inst✝² : CompleteSpace S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
f : R⟦X⟧
this : HasSum (fun d ↦ φ ((MvPowerSeries.coeff d) f) * d.prod fun s e ↦ a ^ e) (MvPowerSeries.eval₂ φ (fun x ↦ a) f)
⊢ ∀ x ∉ Set.range (Finsupp.single ()), (φ ((MvPowerSeries.coeff x) f) * x.prod fun s e ↦ a ^ e) = 0

Tactic:
intro d hd

Goal state after:
R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
φ : R →+* S
a : S
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformSpace S
inst✝⁶ : IsUniformAddGroup R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : T2Space S
inst✝² : CompleteSpace S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
f : R⟦X⟧
this : HasSum (fun d ↦ φ ((MvPowerSeries.coeff d) f) * d.prod fun s e ↦ a ^ e) (MvPowerSeries.eval₂ φ (fun x ↦ a) f)
d : Unit →₀ ℕ
hd : d ∉ Set.range (Finsupp.single ())
⊢ (φ ((MvPowerSeries.coeff d) f) * d.prod fun s e ↦ a ^ e) = 0
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
⊢ ∀ (x : σ →₀ ℕ), x ≠ e → (coeff x) f • (coeff e) (x.prod fun s e ↦ X s ^ e) = 0

Tactic:
intro d hd

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 d : σ →₀ ℕ
hd : d ≠ e
⊢ (coeff d) f • (coeff e) (d.prod fun s e ↦ X s ^ e) = 0
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
f : MvPowerSeries σ R
n : τ →₀ ℕ
⊢ ∀ (x : σ →₀ ℕ), x ≠ 0 → (coeff x) f • (coeff n) (x.prod fun s e ↦ 0 s ^ e) = 0

Tactic:
intro d hd

Goal state after:
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
f : MvPowerSeries σ R
n : τ →₀ ℕ
d : σ →₀ ℕ
hd : d ≠ 0
⊢ (coeff d) f • (coeff n) (d.prod fun s e ↦ 0 s ^ e) = 0
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
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
⊢ ∀ (d : τ →₀ ℕ),
    ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d →
      (coeff d) (subst a f) = 0

Tactic:
intro d hd

Goal state after:
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
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
d : τ →₀ ℕ
hd : ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d
⊢ (coeff d) (subst a f) = 0