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

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x x' : L
⊢ ‖x‖ < ‖x'‖ ↔ (RankOne.hom v) (v.restrict x) < (RankOne.hom v) (v.restrict x')

Tactic:
]

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x x' : L
⊢ ‖x‖ < ‖x'‖ ↔ (RankOne.hom v) (v.restrict x) < (RankOne.hom v) (v.restrict x')
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: case inr
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f < ⊤
hg : weightedOrder w g < ⊤
p : ℕ := (weightedOrder w f).toNat
hp : ↑p = weightedOrder w f
q : ℕ := (weightedOrder w g).toNat
hq : ↑q = weightedOrder w g
H : (weightedHomogeneousComponent w q) g = 0
⊢ ↑(weightedOrder w g).toNat = weightedOrder w g

Tactic:
]

Goal state after:
case inr
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f < ⊤
hg : weightedOrder w g < ⊤
p : ℕ := (weightedOrder w f).toNat
hp : ↑p = weightedOrder w f
q : ℕ := (weightedOrder w g).toNat
hq : ↑q = weightedOrder w g
H : (weightedHomogeneousComponent w q) g = 0
⊢ ↑(weightedOrder w g).toNat = weightedOrder w g
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
⊢ ‖x‖ ≤ 1 ↔ v.restrict x ≤ v.restrict 1

Tactic:
]

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
⊢ ‖x‖ ≤ 1 ↔ v.restrict x ≤ v.restrict 1
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
⊢ ‖x‖ < 1 ↔ v x < v 1

Tactic:
← v.restrict_lt_iff

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
⊢ ‖x‖ < 1 ↔ v.restrict x < v.restrict 1
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
⊢ ‖x‖ < 1 ↔ v.restrict x < v.restrict 1

Tactic:
]

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
⊢ ‖x‖ < 1 ↔ v.restrict x < v.restrict 1
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: σ : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
f : R →+* S
inst✝ : IsLocalHom f
φ : MvPowerSeries σ R
ψ : (MvPowerSeries σ S)ˣ
h : constantCoeff ↑ψ = f (constantCoeff φ)
⊢ IsUnit φ

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
f : R →+* S
inst✝ : IsLocalHom f
φ : MvPowerSeries σ R
ψ : (MvPowerSeries σ S)ˣ
h : constantCoeff ↑ψ = f (constantCoeff φ)
⊢ IsUnit φ
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
⊢ 1 ≤ ‖x‖ ↔ v.restrict 1 ≤ v.restrict x

Tactic:
]

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
⊢ 1 ≤ ‖x‖ ↔ v.restrict 1 ≤ v.restrict x
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: σ : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
f : R →+* S
inst✝ : IsLocalHom f
φ : MvPowerSeries σ R
ψ : (MvPowerSeries σ S)ˣ
h : constantCoeff ↑ψ = f (constantCoeff φ)
this : IsUnit (f (constantCoeff φ))
⊢ IsUnit φ

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
f : R →+* S
inst✝ : IsLocalHom f
φ : MvPowerSeries σ R
ψ : (MvPowerSeries σ S)ˣ
h : constantCoeff ↑ψ = f (constantCoeff φ)
this : IsUnit (f (constantCoeff φ))
⊢ IsUnit φ
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
⊢ 1 < ‖x‖ ↔ v 1 < v x

Tactic:
← v.restrict_lt_iff

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
⊢ 1 < ‖x‖ ↔ v.restrict 1 < v.restrict x
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
⊢ 1 < ‖x‖ ↔ v.restrict 1 < v.restrict x

Tactic:
]

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x : L
⊢ 1 < ‖x‖ ↔ v.restrict 1 < v.restrict x
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: σ : Type u_1
k : Type u_3
inst✝ : Field k
φ : MvPowerSeries σ k
⊢ (constantCoeff φ)⁻¹ = (constantCoeff φ)⁻¹

Tactic:
]

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

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f < ⊤
hg : weightedOrder w g < ⊤
p : ℕ := (weightedOrder w f).toNat
hp : ↑p = weightedOrder w f
q : ℕ := (weightedOrder w g).toNat
hq : ↑q = weightedOrder w g
this : (weightedHomogeneousComponent w (p + q)) (f * g) ≠ 0
⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g

Tactic:
]

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f < ⊤
hg : weightedOrder w g < ⊤
p : ℕ := (weightedOrder w f).toNat
hp : ↑p = weightedOrder w f
q : ℕ := (weightedOrder w g).toNat
hq : ↑q = weightedOrder w g
this : (weightedHomogeneousComponent w (p + q)) (f * g) ≠ 0
⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: σ : Type u_1
k : Type u_3
inst✝ : Field k
φ : MvPowerSeries σ k
h : constantCoeff φ = 0
n : σ →₀ ℕ
⊢ (if n = 0 then (constantCoeff φ)⁻¹
    else -(constantCoeff φ)⁻¹ * ∑ x ∈ antidiagonal n, if x.2 < n then (coeff x.1) φ * (coeff x.2) φ⁻¹ else 0) =
    (coeff n) 0

Tactic:
]

Goal state after:
σ : Type u_1
k : Type u_3
inst✝ : Field k
φ : MvPowerSeries σ k
h : constantCoeff φ = 0
n : σ →₀ ℕ
⊢ (if n = 0 then (constantCoeff φ)⁻¹
    else -(constantCoeff φ)⁻¹ * ∑ x ∈ antidiagonal n, if x.2 < n then (coeff x.1) φ * (coeff x.2) φ⁻¹ else 0) =
    (coeff n) 0
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: σ : Type u_1
k : Type u_3
inst✝ : Field k
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: σ : Type u_1
k : Type u_3
inst✝ : Field k
φ : MvPowerSeries σ k
u : kˣ
h : constantCoeff φ = ↑u
⊢ φ.invOfUnit u = φ.invOfUnit (Units.mk0 (constantCoeff φ) ⋯)

Tactic:
]

Goal state after:
σ : Type u_1
k : Type u_3
inst✝ : Field k
φ : MvPowerSeries σ k
u : kˣ
h : constantCoeff φ = ↑u
⊢ φ.invOfUnit u = φ.invOfUnit (Units.mk0 (constantCoeff φ) ⋯)
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: σ : Type u_1
k : Type u_3
inst✝ : Field k
φ : MvPowerSeries σ k
u : kˣ
h : constantCoeff φ = ↑u
⊢ ↑u = ↑(Units.mk0 (constantCoeff φ) ⋯)

Tactic:
]

Goal state after:
σ : Type u_1
k : Type u_3
inst✝ : Field k
φ : MvPowerSeries σ k
u : kˣ
h : constantCoeff φ = ↑u
⊢ ↑u = ↑(Units.mk0 (constantCoeff φ) ⋯)
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: σ : Type u_1
k : Type u_3
inst✝ : Field k
φ : MvPowerSeries σ k
h : constantCoeff φ ≠ 0
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f < ⊤
hg : weightedOrder w g < ⊤
p : ℕ := (weightedOrder w f).toNat
hp : ↑p = weightedOrder w f
q : ℕ := (weightedOrder w g).toNat
hq : ↑q = weightedOrder w g
this : (weightedHomogeneousComponent w (p + q)) (f * g) ≠ 0
⊢ ¬↑(p + q) < weightedOrder w (f * g)

Tactic:
]

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f < ⊤
hg : weightedOrder w g < ⊤
p : ℕ := (weightedOrder w f).toNat
hp : ↑p = weightedOrder w f
q : ℕ := (weightedOrder w g).toNat
hq : ↑q = weightedOrder w g
this : (weightedHomogeneousComponent w (p + q)) (f * g) ≠ 0
⊢ ¬↑(p + q) < weightedOrder w (f * g)
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: σ : Type u_1
k : Type u_3
inst✝ : Field k
φ ψ : MvPowerSeries σ k
h : constantCoeff ψ ≠ 0
⊢ φ = ψ⁻¹ ↔ φ = ψ⁻¹

Tactic:
]

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

Goal state before:
pp: σ : Type u_1
k : Type u_3
inst✝ : Field k
φ ψ : MvPowerSeries σ k
h : constantCoeff ψ ≠ 0
⊢ φ * ψ = 1 ↔ φ * ψ = 1

Tactic:
]

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

Goal state before:
pp: case pos
σ : Type u_1
k : Type u_3
inst✝ : Field k
φ ψ : MvPowerSeries σ k
h : constantCoeff (φ * ψ) = 0
⊢ 0 = ψ⁻¹ * φ⁻¹

Tactic:
]

Goal state after:
case pos
σ : Type u_1
k : Type u_3
inst✝ : Field k
φ ψ : MvPowerSeries σ k
h : constantCoeff (φ * ψ) = 0
⊢ 0 = ψ⁻¹ * φ⁻¹
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: case neg
σ : Type u_1
k : Type u_3
inst✝ : Field k
φ ψ : MvPowerSeries σ k
h : ¬constantCoeff (φ * ψ) = 0
⊢ ψ⁻¹ * φ⁻¹ * (φ * ψ) = 1

Tactic:
]

Goal state after:
case neg
σ : Type u_1
k : Type u_3
inst✝ : Field k
φ ψ : MvPowerSeries σ k
h : ¬constantCoeff (φ * ψ) = 0
⊢ ψ⁻¹ * φ⁻¹ * (φ * ψ) = 1
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f < ⊤
hg : weightedOrder w g = ⊤
⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g

Tactic:
]

Goal state after:
case neg
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f < ⊤
hg : weightedOrder w g = ⊤
⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g
Module: Mathlib.RingTheory.PowerSeries.NoZeroDivisors

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
this : Ideal.span {X} = RingHom.ker constantCoeff
⊢ (RingHom.ker constantCoeff).IsPrime

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
this : Ideal.span {X} = RingHom.ker constantCoeff
⊢ (RingHom.ker constantCoeff).IsPrime
Module: Mathlib.RingTheory.PowerSeries.NoZeroDivisors

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
φ : R⟦X⟧
⊢ constantCoeff φ = 0 ↔ constantCoeff φ = 0

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
⊢ (Ideal.span {X}).IsPrime

pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
⊢ X ≠ 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
⊢ (Ideal.span {X}).IsPrime

R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
⊢ X ≠ 0
Module: Mathlib.RingTheory.PowerSeries.NoZeroDivisors

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a : R
ha : a ≠ 0
p q : R⟦X⟧
h : ∀ (n : ℕ), (coeff n) ((rescale a) p) = (coeff n) ((rescale a) q)
⊢ ∀ (n : ℕ), (coeff n) p = (coeff n) q

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a : R
ha : a ≠ 0
p q : R⟦X⟧
h : ∀ (n : ℕ), (coeff n) ((rescale a) p) = (coeff n) ((rescale a) q)
⊢ ∀ (n : ℕ), (coeff n) p = (coeff n) q
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: case neg
σ : Type u_1
k : Type u_3
inst✝ : Field k
φ ψ : MvPowerSeries σ k
h : ¬constantCoeff φ = 0 ∧ ¬constantCoeff ψ = 0
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
k : Type u_3
inst✝ : Field k
⊢ 1 = 1

pp: σ : Type u_1
R : Type u_2
k : Type u_3
inst✝ : Field k
⊢ constantCoeff 1 ≠ 0

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
k : Type u_3
inst✝ : Field k
⊢ constantCoeff 1 ≠ 0
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f = ⊤
⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g

Tactic:
]

Goal state after:
case neg
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f = ⊤
⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g
Module: Mathlib.RingTheory.PowerSeries.NoZeroDivisors

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a : R
ha : a ≠ 0
p q : R⟦X⟧
n : ℕ
h : (coeff n) p = (coeff n) q
⊢ (coeff n) p = (coeff n) q

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a : R
ha : a ≠ 0
p q : R⟦X⟧
n : ℕ
h : (coeff n) p = (coeff n) q
⊢ (coeff n) p = (coeff n) q
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
n : ℕ
a : R
φ : R⟦X⟧
⊢ (if single () n = 0 then a
    else
      -a *
        ∑ x ∈ antidiagonal (single () n),
          if x.2 < single () n then (MvPowerSeries.coeff x.1) φ * (MvPowerSeries.coeff x.2) (MvPowerSeries.inv.aux a φ)
          else 0) =
    if n = 0 then a
    else -a * ∑ x ∈ antidiagonal n, if x.2 < n then (coeff x.1) φ * (coeff x.2) (MvPowerSeries.inv.aux a φ) else 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : Ring R
n : ℕ
a : R
φ : R⟦X⟧
⊢ (if single () n = 0 then a
    else
      -a *
        ∑ x ∈ antidiagonal (single () n),
          if x.2 < single () n then (MvPowerSeries.coeff x.1) φ * (MvPowerSeries.coeff x.2) (MvPowerSeries.inv.aux a φ)
          else 0) =
    if n = 0 then a
    else -a * ∑ x ∈ antidiagonal n, if x.2 < n then (coeff x.1) φ * (coeff x.2) (MvPowerSeries.inv.aux a φ) else 0
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: case inr
σ : Type u_1
k : Type u_3
inst✝ : Field k
r : k
hr : r ≠ 0
⊢ 1 = 1

pp: case inr
σ : Type u_1
k : Type u_3
inst✝ : Field k
r : k
hr : r ≠ 0
⊢ constantCoeff (C r) ≠ 0

Tactic:
]

Goal state after:
case inr
σ : Type u_1
k : Type u_3
inst✝ : Field k
r : k
hr : r ≠ 0
⊢ constantCoeff (C r) ≠ 0
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: σ : Type u_1
k : Type u_3
inst✝ : Field k
s : σ
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case cons
σ : Type u_1
R : Type u_3
inst✝² : CommSemiring R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
ι : Type u_4
w : σ → ℕ
f : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : weightedOrder w (∏ i ∈ s, f i) = ∑ i ∈ s, weightedOrder w (f i)
⊢ weightedOrder w (f a) + ∑ i ∈ s, weightedOrder w (f i) = weightedOrder w (f a) + ∑ x ∈ s, weightedOrder w (f x)

Tactic:
]

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

Goal state before:
pp: case neg.e_a.h.inr
R : Type u_1
inst✝ : Ring R
n : ℕ
a : R
φ : R⟦X⟧
h✝ : ¬n = 0
i j : ℕ
_hij : (i, j) ∈ antidiagonal n
H : j < n
⊢ (coeff (i, j).1) φ * (coeff (i, j).2) (MvPowerSeries.inv.aux a φ) =
    (MvPowerSeries.coeff
          (match (i, j) with
            | (a, b) => (single () a, single () b)).1)
        φ *
      (MvPowerSeries.coeff
          (match (i, j) with
            | (a, b) => (single () a, single () b)).2)
        (MvPowerSeries.inv.aux a φ)

pp: case neg.e_a.h.inr.hc
R : Type u_1
inst✝ : Ring R
n : ℕ
a : R
φ : R⟦X⟧
h✝ : ¬n = 0
i j : ℕ
_hij : (i, j) ∈ antidiagonal n
H : j < n
⊢ (match (i, j) with
      | (a, b) => (single () a, single () b)).2 <
    single () n

Tactic:
]

Goal state after:
case neg.e_a.h.inr
R : Type u_1
inst✝ : Ring R
n : ℕ
a : R
φ : R⟦X⟧
h✝ : ¬n = 0
i j : ℕ
_hij : (i, j) ∈ antidiagonal n
H : j < n
⊢ (coeff (i, j).1) φ * (coeff (i, j).2) (MvPowerSeries.inv.aux a φ) =
    (MvPowerSeries.coeff
          (match (i, j) with
            | (a, b) => (single () a, single () b)).1)
        φ *
      (MvPowerSeries.coeff
          (match (i, j) with
            | (a, b) => (single () a, single () b)).2)
        (MvPowerSeries.inv.aux a φ)

case neg.e_a.h.inr.hc
R : Type u_1
inst✝ : Ring R
n : ℕ
a : R
φ : R⟦X⟧
h✝ : ¬n = 0
i j : ℕ
_hij : (i, j) ∈ antidiagonal n
H : j < n
⊢ (match (i, j) with
      | (a, b) => (single () a, single () b)).2 <
    single () n
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
φ : R⟦X⟧
u : Rˣ
⊢ ↑u⁻¹ = ↑u⁻¹

Tactic:
]

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

Goal state before:
pp: k : Type u_2
inst✝ : Field k
n : ℕ
φ : k⟦X⟧
⊢ (if n = 0 then (constantCoeff φ)⁻¹
    else
      -(constantCoeff φ)⁻¹ *
        ∑ x ∈ antidiagonal n, if x.2 < n then (coeff x.1) φ * (coeff x.2) (inv.aux (constantCoeff φ)⁻¹ φ) else 0) =
    if n = 0 then (constantCoeff φ)⁻¹
    else
      -(constantCoeff φ)⁻¹ *
        ∑ x ∈ antidiagonal n, if x.2 < n then (coeff x.1) φ * (coeff x.2) (inv.aux (constantCoeff φ)⁻¹ φ) else 0

Tactic:
]

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

Goal state before:
pp: k : Type u_2
inst✝ : Field k
f : k⟦X⟧
hf : f ≠ 0
⊢ f.divXPowOrder * Inv_divided_by_X_pow_order hf = 1

Tactic:
]

Goal state after:
k : Type u_2
inst✝ : Field k
f : k⟦X⟧
hf : f ≠ 0
⊢ f.divXPowOrder * Inv_divided_by_X_pow_order hf = 1
Module: Mathlib.NumberTheory.NumberField.Completion.FinitePlace

Goal state before:
pp: A : Type u_1
inst✝⁴ : CommRing A
inst✝³ : IsDedekindDomain A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
v : HeightOneSpectrum A
hv : Finite (A ⧸ v.asIdeal)
⊢ ¬(Set.range ⇑(valuation K v)).Subsingleton

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁴ : CommRing A
inst✝³ : IsDedekindDomain A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
v : HeightOneSpectrum A
hv : Finite (A ⧸ v.asIdeal)
⊢ ¬(Set.range ⇑(valuation K v)).Subsingleton
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: k : Type u_2
inst✝ : Field k
f : k⟦X⟧
hf : f ≠ 0
h : f = f.divXPowOrder
⊢ IsUnit f.divXPowOrder

Tactic:
]

Goal state after:
k : Type u_2
inst✝ : Field k
f : k⟦X⟧
hf : f ≠ 0
h : f = f.divXPowOrder
⊢ IsUnit f.divXPowOrder
Module: Mathlib.NumberTheory.NumberField.Completion.FinitePlace

Goal state before:
pp: A : Type u_1
inst✝⁴ : CommRing A
inst✝³ : IsDedekindDomain A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
v : HeightOneSpectrum A
hv : Finite (A ⧸ v.asIdeal)
⊢ ¬(Set.range ⇑Valued.v).Subsingleton

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁴ : CommRing A
inst✝³ : IsDedekindDomain A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
v : HeightOneSpectrum A
hv : Finite (A ⧸ v.asIdeal)
⊢ ¬(Set.range ⇑Valued.v).Subsingleton
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: k : Type u_2
inst✝ : Field k
⊢ 1 ∉ Ideal.span {X} ∧ ∀ (J : Ideal k⟦X⟧) (x : k⟦X⟧), Ideal.span {X} ≤ J → x ∉ Ideal.span {X} → x ∈ J → 1 ∈ J

Tactic:
]

Goal state after:
k : Type u_2
inst✝ : Field k
⊢ 1 ∉ Ideal.span {X} ∧ ∀ (J : Ideal k⟦X⟧) (x : k⟦X⟧), Ideal.span {X} ≤ J → x ∉ Ideal.span {X} → x ∈ J → 1 ∈ J
Module: Mathlib.NumberTheory.NumberField.Completion.FinitePlace

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
R : Type u_2
inst✝³ : CommRing R
inst✝² : Algebra R K
inst✝¹ : IsDedekindDomain R
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
h : (valuation K v).IsRankOneDiscrete
⊢ (MonoidWithZeroHom.ofClass (valuation K v)).valueGroup = (MonoidWithZeroHom.ofClass (valuation K v)).valueGroup

Tactic:
]

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

Goal state before:
pp: case left
k : Type u_2
inst✝ : Field k
⊢ ¬X ∣ 1

Tactic:
]

Goal state after:
case left
k : Type u_2
inst✝ : Field k
⊢ ¬X ∣ 1
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: case right
k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
⊢ Ideal.span {X} ≤ I → f ∉ Ideal.span {X} → f ∈ I → 1 ∈ I

Tactic:
hI

Goal state after:
case right
k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
hI : Ideal.span {X} ≤ I
⊢ f ∉ Ideal.span {X} → f ∈ I → 1 ∈ I
Module: Mathlib.NumberTheory.NumberField.Completion.FinitePlace

Goal state before:
pp: R : Type u_2
inst✝³ : CommRing R
inst✝² : IsDedekindDomain R
v : HeightOneSpectrum R
inst✝¹ : Module.Finite ℤ R
inst✝ : Module.Free ℤ R
h : absNorm v.asIdeal ≤ 1
⊢ absNorm v.asIdeal = 1

Tactic:
]

Goal state after:
R : Type u_2
inst✝³ : CommRing R
inst✝² : IsDedekindDomain R
v : HeightOneSpectrum R
inst✝¹ : Module.Finite ℤ R
inst✝ : Module.Free ℤ R
h : absNorm v.asIdeal ≤ 1
⊢ absNorm v.asIdeal = 1
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: case right
k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
hI : Ideal.span {X} ≤ I
hfX : ¬constantCoeff f = 0
hfI : f ∈ I
⊢ 1 ∈ I

Tactic:
]

Goal state after:
case right
k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
hI : Ideal.span {X} ≤ I
hfX : ¬constantCoeff f = 0
hfI : f ∈ I
⊢ 1 ∈ I
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
hI : Ideal.span {X} ≤ I
hfX : ¬constantCoeff f = 0
hfI : f ∈ I
⊢ C (f 0) = C (f 0)

Tactic:
]

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

Goal state before:
pp: k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
hI : Ideal.span {X} ≤ I
hfX : ¬constantCoeff f = 0
hfI : f ∈ I
this : C (f 0) = f - (f - C (f 0))
⊢ f - (f - C (f 0)) ∈ I

Tactic:
]

Goal state after:
k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
hI : Ideal.span {X} ≤ I
hfX : ¬constantCoeff f = 0
hfI : f ∈ I
this : C (f 0) = f - (f - C (f 0))
⊢ f - (f - C (f 0)) ∈ I
Module: Mathlib.NumberTheory.NumberField.Completion.FinitePlace

Goal state before:
pp: R : Type u_2
inst✝³ : CommRing R
inst✝² : IsDedekindDomain R
v : HeightOneSpectrum R
inst✝¹ : Module.Finite ℤ R
inst✝ : Module.Free ℤ R
h : absNorm v.asIdeal ≤ 1
⊢ Finite (R ⧸ v.asIdeal)

Tactic:
]

Goal state after:
R : Type u_2
inst✝³ : CommRing R
inst✝² : IsDedekindDomain R
v : HeightOneSpectrum R
inst✝¹ : Module.Finite ℤ R
inst✝ : Module.Free ℤ R
h : absNorm v.asIdeal ≤ 1
⊢ Finite (R ⧸ v.asIdeal)
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
hI : Ideal.span {X} ≤ I
hfX : ¬constantCoeff f = 0
hfI : f ∈ I
this : C (f 0) = f - (f - C (f 0))
⊢ constantCoeff f = f 0

Tactic:
]

Goal state after:
k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
hI : Ideal.span {X} ≤ I
hfX : ¬constantCoeff f = 0
hfI : f ∈ I
this : C (f 0) = f - (f - C (f 0))
⊢ constantCoeff f = f 0
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: case right
k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
hI : Ideal.span {X} ≤ I
hfX : ¬constantCoeff f = 0
hfI : f ∈ I
hfI0 : C (f 0) ∈ I
⊢ I = ⊤

Tactic:
]

Goal state after:
case right
k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
hI : Ideal.span {X} ≤ I
hfX : ¬constantCoeff f = 0
hfI : f ∈ I
hfI0 : C (f 0) ∈ I
⊢ I = ⊤
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: k : Type u_2
inst✝ : Field k
hX : (Ideal.span {X}).IsMaximal
⊢ IsLocalRing.maximalIdeal k⟦X⟧ = IsLocalRing.maximalIdeal k⟦X⟧

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Completion.FinitePlace

Goal state before:
pp: K : Type u_1
inst✝⁶ : Field K
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R K
inst✝³ : IsDedekindDomain R
inst✝² : IsFractionRing R K
v : HeightOneSpectrum R
inst✝¹ : Module.Finite ℤ R
inst✝ : Module.Free ℤ R
x : K
⊢ ↑((toNNReal ⋯) ((valuation K v) x)) = ↑((toNNReal ⋯) ((valuation K v) x))

Tactic:
]

Goal state after: