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

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0
m : σ →₀ ℕ
H : ¬m - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
⊢ single s n + (i, j).2 - single s n + single s n = single s n + (i, j).2

Tactic:
]

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0
m : σ →₀ ℕ
H : ¬m - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
⊢ single s n + (i, j).2 - single s n + single s n = single s n + (i, j).2
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
⊢ r ∈ v.asIdeal ↔ r ∈ v.asIdeal

Tactic:
]

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
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0
m : σ →₀ ℕ
H : ¬m - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
a✝ : σ
⊢ (single s n) a✝ + (i, j).2 a✝ = (single s n) a✝ + (i, j).2 a✝

Tactic:
]

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

Goal state before:
pp: case neg
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 : embedding a ≠ embedding 0
h0 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h0' : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : v x = v y
⊢ v x = v y

Tactic:
]

Goal state after:
case neg
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 : embedding a ≠ embedding 0
h0 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h0' : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : v x = v y
⊢ v x = v y
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: R : Type u_2
inst✝ : CommSemiring R
φ ψ : R⟦X⟧
this : antidiagonal 1 = {(0, 1), (1, 0)}
⊢ (coeff (1, 0).1) φ * constantCoeff ψ + (coeff (0, 1).2) ψ * constantCoeff φ =
    (coeff 1) φ * constantCoeff ψ + (coeff 1) ψ * constantCoeff φ

pp: R : Type u_2
inst✝ : CommSemiring R
φ ψ : R⟦X⟧
this : antidiagonal 1 = {(0, 1), (1, 0)}
⊢ (0, 1) ∉ {(1, 0)}

Tactic:
]

Goal state after:
R : Type u_2
inst✝ : CommSemiring R
φ ψ : R⟦X⟧
this : antidiagonal 1 = {(0, 1), (1, 0)}
⊢ (0, 1) ∉ {(1, 0)}
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
φ : MvPowerSeries σ R
⊢ (∀ (m : σ →₀ ℕ), m s < 1 → (coeff m) φ = 0) ↔ ∀ (m : σ →₀ ℕ), m s = 0 → (coeff m) φ = 0

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
φ : MvPowerSeries σ R
⊢ (∀ (m : σ →₀ ℕ), m s < 1 → (coeff m) φ = 0) ↔ ∀ (m : σ →₀ ℕ), m s = 0 → (coeff m) φ = 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
hr : ¬r = 0
⊢ n ≤ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r})).factors ↔
    n ≤ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r})).factors

Tactic:
]

Goal state after:
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
n : ℕ
⊢ r ∈ v.asIdeal ^ n ↔ r ∈ v.asIdeal ^ n

Tactic:
]

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

Goal state before:
pp: case mp
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < 1 → (coeff m) φ = 0
m : σ →₀ ℕ
hm : m s = 0
⊢ (coeff m) φ = 0

Tactic:
exact h m (hm.symm ▸ zero_lt_one)

Goal state after:
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
n : ℕ
⊢ ↑n ≤ emultiplicity v.asIdeal (Ideal.span {r}) ↔ ↑n ≤ emultiplicity v.asIdeal (Ideal.span {r})

Tactic:
]

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
⊢ ∑ p ∈ antidiagonal d, (coeff p.1) (f a) * (coeff p.2) (∏ x ∈ s, f x) =
    ∑ x ∈ antidiagonal d,
      ∑ i ∈ Finset.map { toFun := fun f ↦ (↑f).update a x.1, inj' := ⋯ } (s.finsuppAntidiag x.2).attach,
        ∏ i_1 ∈ insert a s, (coeff (i i_1)) (f i_1)

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:
]

Goal state after:
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
⊢ ∑ p ∈ antidiagonal d, (coeff p.1) (f a) * (coeff p.2) (∏ x ∈ s, f x) =
    ∑ x ∈ antidiagonal d,
      ∑ i ∈ Finset.map { toFun := fun f ↦ (↑f).update a x.1, inj' := ⋯ } (s.finsuppAntidiag x.2).attach,
        ∏ i_1 ∈ insert a s, (coeff (i i_1)) (f i_1)

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
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
n : ℕ
⊢ exp (-↑n) ≤ v.intValuation r ↔ exp (-↑n) < v.intValuation r * exp 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
⊢ exp (-↑n) ≤ v.intValuation r ↔ exp (-↑n) < v.intValuation r * exp 1
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
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding) γ)}
⊢ (∃ i, True ∧ {x | extensionValuation x < ↑((Units.map ↑embedding) i)} ⊆ s) ↔
    ∃ γ, {x | extensionValuation.restrict x < ↑γ} ⊆ s

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding) γ)}
⊢ (∃ i, True ∧ {x | extensionValuation x < ↑((Units.map ↑embedding) i)} ⊆ s) ↔
    ∃ γ, {x | extensionValuation.restrict x < ↑γ} ⊆ s
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case inr.succ
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
⊢ (coeff (0, 1).1) (φ ^ n') * (coeff (0, 1).2) φ + (coeff (1, 0).1) (φ ^ n') * (coeff (1, 0).2) φ =
    ↑(n' + 1) * (coeff 1) φ * constantCoeff φ ^ (n' + 1 - 1)

pp: case inr.succ
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
⊢ (0, 1) ∉ {(1, 0)}

Tactic:
]

Goal state after:
case inr.succ
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
⊢ (coeff (0, 1).1) (φ ^ n') * (coeff (0, 1).2) φ + (coeff (1, 0).1) (φ ^ n') * (coeff (1, 0).2) φ =
    ↑(n' + 1) * (coeff 1) φ * constantCoeff φ ^ (n' + 1 - 1)

case inr.succ
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
⊢ (0, 1) ∉ {(1, 0)}
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
hv : ¬v.intValuation r = 0
⊢ exp (-↑n) ≤ v.intValuation r ↔ exp (-↑n) ≤ v.intValuation r

Tactic:
]

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)
u v : σ →₀ ℕ
⊢ ∑ x ∈ (s.finsuppAntidiag v).attach, (coeff u) (f a) * ∏ i ∈ s, (coeff (↑x i)) (f i) =
    ∑ x ∈ (s.finsuppAntidiag v).attach, ∏ x_1 ∈ insert a s, (coeff (Function.update (⇑↑x) a u x_1)) (f x_1)

Tactic:
]

Goal state after:
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)
u v : σ →₀ ℕ
⊢ ∑ x ∈ (s.finsuppAntidiag v).attach, (coeff u) (f a) * ∏ i ∈ s, (coeff (↑x i)) (f i) =
    ∑ x ∈ (s.finsuppAntidiag v).attach, ∏ x_1 ∈ insert a s, (coeff (Function.update (⇑↑x) a u x_1)) (f x_1)
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
⊢ DvdNotUnit v.asIdeal (v.asIdeal ^ 2)

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
⊢ DvdNotUnit v.asIdeal (v.asIdeal ^ 2)
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)
u v : σ →₀ ℕ
x : ι →₀ σ →₀ ℕ
⊢ (coeff u) (f a) * s.prod ?m.163 = (coeff u) (f a) * ∏ x_1 ∈ s, (coeff (Function.update (⇑x) a u x_1)) (f x_1)

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)
u v : σ →₀ ℕ
x : ι →₀ σ →₀ ℕ
⊢ ∀ x_1 ∈ s, (coeff (x x_1)) (f x_1) = ?m.163 x_1

pp: 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)
u v : σ →₀ ℕ
x : ι →₀ σ →₀ ℕ
⊢ ι → R

Tactic:
]

Goal state after:
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)
u v : σ →₀ ℕ
x : ι →₀ σ →₀ ℕ
⊢ ∀ x_1 ∈ s, (coeff (x x_1)) (f x_1) = (coeff (Function.update (⇑x) a u x_1)) (f x_1)
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
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding) γ)}
x : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
γ : (ofClass v).ValueGroup₀ˣ
⊢ (if h : (ofClass extensionValuation) x = 0 then 0 else ↑⟨Units.mk0 ((ofClass extensionValuation) x) h, ⋯⟩) <
      valueGroup₀_equiv_extensionValuation ↑γ ↔
    extension x < ↑γ

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding) γ)}
x : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
γ : (ofClass v).ValueGroup₀ˣ
⊢ (if h : (ofClass extensionValuation) x = 0 then 0 else ↑⟨Units.mk0 ((ofClass extensionValuation) x) h, ⋯⟩) <
      valueGroup₀_equiv_extensionValuation ↑γ ↔
    extension x < ↑γ
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : π ∈ v.asIdeal
notMem : π ∉ v.asIdeal ^ 2
⊢ π ≠ 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : π ∈ v.asIdeal
notMem : π ∉ v.asIdeal ^ 2
⊢ π ≠ 0
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
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding) γ)}
x : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
γ : (ofClass v).ValueGroup₀ˣ
⊢ extensionValuation.restrict x < valueGroup₀_equiv_extensionValuation ↑γ ↔ embedding (extension x) < embedding ↑γ

Tactic:
rw [embedding_strictMono.lt_iff_lt, Valuation.restrict_def, restrict₀_apply]

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding) γ)}
x : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
γ : (ofClass v).ValueGroup₀ˣ
⊢ (if h : (ofClass extensionValuation) x = 0 then 0 else ↑⟨Units.mk0 ((ofClass extensionValuation) x) h, ⋯⟩) <
      valueGroup₀_equiv_extensionValuation ↑γ ↔
    extension x < ↑γ
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)
u v : σ →₀ ℕ
x : ι →₀ σ →₀ ℕ
i : ι
hi : i ∈ s
⊢ (coeff (x i)) (f i) = (coeff (x i)) (f i)

pp: case insert.h
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)
u v : σ →₀ ℕ
x : ι →₀ σ →₀ ℕ
i : ι
hi : i ∈ s
⊢ i ≠ a

Tactic:
]

Goal state after:
case insert.h
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)
u v : σ →₀ ℕ
x : ι →₀ σ →₀ ℕ
i : ι
hi : i ∈ s
⊢ i ≠ a
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : π ∈ v.asIdeal
notMem : 0 ∉ v.asIdeal ^ 2
h : π = 0
⊢ False

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : π ∈ v.asIdeal
notMem : 0 ∉ v.asIdeal ^ 2
h : π = 0
⊢ False
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case inr.succ.inl
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
h' : n' = 0
h'' :
  ¬constantCoeff φ ^ 0 * (coeff 1) φ + (coeff 1) (φ ^ 0) * constantCoeff φ =
      (↑0 + 1) * (coeff 1) φ * constantCoeff φ ^ 0
⊢ False

Tactic:
]

Goal state after:
case inr.succ.inl
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
h' : n' = 0
h'' :
  ¬constantCoeff φ ^ 0 * (coeff 1) φ + (coeff 1) (φ ^ 0) * constantCoeff φ =
      (↑0 + 1) * (coeff 1) φ * constantCoeff φ ^ 0
⊢ False
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : π ∈ v.asIdeal
notMem : π ∉ v.asIdeal ^ 2
hπ : Associates.mk (Ideal.span {π}) ≠ 0
⊢ -↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors) = -1

Tactic:
]

Goal state after:
case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : π ∈ v.asIdeal
notMem : π ∉ v.asIdeal ^ 2
hπ : Associates.mk (Ideal.span {π}) ≠ 0
⊢ -↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors) = -1
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
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding) γ)}
x : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
γ : (ofClass v).ValueGroup₀ˣ
hx0 : x = 0
⊢ 0 < valueGroup₀_equiv_extensionValuation ↑γ ↔ extension 0 < ↑γ

Tactic:
]

Goal state after:
case pos
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding) γ)}
x : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
γ : (ofClass v).ValueGroup₀ˣ
hx0 : x = 0
⊢ 0 < valueGroup₀_equiv_extensionValuation ↑γ ↔ extension 0 < ↑γ
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : π ∈ v.asIdeal
notMem : π ∉ v.asIdeal ^ 2
hπ : Associates.mk (Ideal.span {π}) ≠ 0
⊢ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors = 1

Tactic:
]

Goal state after:
case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : π ∈ v.asIdeal
notMem : π ∉ v.asIdeal ^ 2
hπ : Associates.mk (Ideal.span {π}) ≠ 0
⊢ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors = 1
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case inr.succ.inr
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
h' : 1 ≤ n'
⊢ constantCoeff φ ^ n' * (coeff 1) φ + ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1) * constantCoeff φ =
    (↑n' + 1) * (coeff 1) φ * constantCoeff φ ^ n'

pp: case inr.succ.inr
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
h' : 1 ≤ n'
⊢ n' > 0

Tactic:
]

Goal state after:
case inr.succ.inr
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
h' : 1 ≤ n'
⊢ constantCoeff φ ^ n' * (coeff 1) φ + ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1) * constantCoeff φ =
    (↑n' + 1) * (coeff 1) φ * constantCoeff φ ^ n'

case inr.succ.inr
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
h' : 1 ≤ n'
⊢ n' > 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : Associates.mk v.asIdeal ≤ Associates.mk (Ideal.span {π})
notMem : ¬Associates.mk (v.asIdeal ^ 2) ≤ Associates.mk (Ideal.span {π})
hπ : Associates.mk (Ideal.span {π}) ≠ 0
⊢ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors = 1

Tactic:
]

Goal state after:
case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : Associates.mk v.asIdeal ≤ Associates.mk (Ideal.span {π})
notMem : ¬Associates.mk (v.asIdeal ^ 2) ≤ Associates.mk (Ideal.span {π})
hπ : Associates.mk (Ideal.span {π}) ≠ 0
⊢ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors = 1