mathlib-initiative/mathlib-tactics huggingface dataset:
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
h : Function.Surjective ⇑v
γ : Γ₀
H : ∃ x, v x = γ
⊢ IsClosed {a | v a = γ → ∃ a, v a = γ}

pp: case neg
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
⊢ IsClosed {a | v a = γ → ∃ a, v a = 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
h : Function.Surjective ⇑v
γ : Γ₀
H : ∃ x, v x = γ
⊢ IsClosed {a | v a = γ → ∃ a, v a = γ}
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
h : Function.Surjective ⇑v
γ : Γ₀
H : ¬∃ x, v x = γ
⊢ IsClosed {a | v a = γ → ∃ a, v a = γ}

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
h : Function.Surjective ⇑v
γ : Γ₀
H : ¬∃ x, v x = γ
⊢ IsClosed {a | v a = γ → ∃ a, v a = γ}
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) a) = ⊤

pp: case neg
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) a) = ↑((weight w) d)

Tactic:
·

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
⊢ weightedOrder w ((monomial d) a) = ⊤
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case neg.inl
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
h : Function.Surjective ⇑v
H : ¬∃ x, v x = 0
⊢ IsClosed {a | ¬v a = 0}

pp: case neg.inr
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
⊢ IsClosed {a | ¬v a = γ}

Tactic:
·

Goal state after:
case neg.inl
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
h : Function.Surjective ⇑v
H : ¬∃ x, v x = 0
⊢ IsClosed {a | ¬v a = 0}
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case neg.inr
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
⊢ IsClosed {a | ¬v a = γ}

Tactic:
·

Goal state after:
case neg.inr
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
⊢ IsClosed {a | ¬v a = γ}
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case neg
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) a) = ↑((weight w) d)

Tactic:
·

Goal state after:
case neg
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) a) = ↑((weight w) d)
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝² : CommSemiring R
A : Type u_4
inst✝¹ : CommSemiring A
inst✝ : Algebra A R
e : σ →₀ ℕ
a : σ → A
⊢ ∀ x ∈ e.support,
    (a x • MvPowerSeries.X x) ^ e x = (MvPowerSeries.C ((algebraMap A R) (a x)) * MvPowerSeries.X x) ^ e x

Tactic:
·

Goal state after:
σ : Type u_1
R : Type u_2
inst✝² : CommSemiring R
A : Type u_4
inst✝¹ : CommSemiring A
inst✝ : Algebra A R
e : σ →₀ ℕ
a : σ → A
⊢ ∀ x ∈ e.support,
    (a x • MvPowerSeries.X x) ^ e x = (MvPowerSeries.C ((algebraMap A R) (a x)) * MvPowerSeries.X x) ^ e x
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case neg.left
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
⊢ ∃ d_1, (coeff d_1) ((monomial d) a) ≠ 0 ∧ (weight w) d_1 = (weight w) d

pp: case neg.right
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
⊢ ∀ (d_1 : σ →₀ ℕ), (weight w) d_1 < (weight w) d → (coeff d_1) ((monomial d) a) = 0

Tactic:
·

Goal state after:
case neg.left
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
⊢ ∃ d_1, (coeff d_1) ((monomial d) a) ≠ 0 ∧ (weight w) d_1 = (weight w) d
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case neg.right
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
⊢ ∀ (d_1 : σ →₀ ℕ), (weight w) d_1 < (weight w) d → (coeff d_1) ((monomial d) a) = 0

Tactic:
·

Goal state after:
case neg.right
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
⊢ ∀ (d_1 : σ →₀ ℕ), (weight w) d_1 < (weight w) d → (coeff d_1) ((monomial d) a) = 0
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case mp.ih
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
a✝ : K
ha : v ↑a✝ = γ
⊢ ∃ a, v a = γ

Tactic:
·

Goal state after:
case mp.ih
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
a✝ : K
ha : v ↑a✝ = γ
⊢ ∃ a, v a = γ
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case mpr
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
a : K
ha : v a = γ
⊢ ∃ a, v a = γ

Tactic:
·

Goal state after:
case mpr
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
a : K
ha : v a = γ
⊢ ∃ a, v a = γ
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case inl
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
n : R
d : ↥v.asIdeal.primeCompl
hnd : x * (algebraMap R K) ↑d = (algebraMap R K) n
⊢ ∃ n d, x * (algebraMap R K) ↑d = (algebraMap R K) n

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
v : HeightOneSpectrum R
x : K
hv : (valuation K v) x ≤ 1
n : R
d : ↥v.asIdeal.primeCompl
hnd : x * (algebraMap R K) n = (algebraMap R K) ↑d
⊢ ∃ n d, x * (algebraMap R K) ↑d = (algebraMap R K) n

Tactic:
·

Goal state after:
case inl
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
n : R
d : ↥v.asIdeal.primeCompl
hnd : x * (algebraMap R K) ↑d = (algebraMap R K) n
⊢ ∃ n d, x * (algebraMap R K) ↑d = (algebraMap R K) n
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
v : HeightOneSpectrum R
x : K
hv : (valuation K v) x ≤ 1
n : R
d : ↥v.asIdeal.primeCompl
hnd : x * (algebraMap R K) n = (algebraMap R K) ↑d
⊢ ∃ n d, x * (algebraMap R K) ↑d = (algebraMap R K) n

Tactic:
·

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
v : HeightOneSpectrum R
x : K
hv : (valuation K v) x ≤ 1
n : R
d : ↥v.asIdeal.primeCompl
hnd : x * (algebraMap R K) n = (algebraMap R K) ↑d
⊢ ∃ n d, x * (algebraMap R K) ↑d = (algebraMap R K) n
Module: Mathlib.Data.Finset.Finsupp

Goal state before:
pp: case refine_1
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι → Finset α
⊢ (∃ a ∈ s.pi t, { toFun := indicator s, inj' := ⋯ } a = f) → f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i

pp: case refine_2
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι → Finset α
⊢ (f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i) → ∃ a ∈ s.pi t, { toFun := indicator s, inj' := ⋯ } a = f

Tactic:
·

Goal state after:
case refine_1
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι → Finset α
⊢ (∃ a ∈ s.pi t, { toFun := indicator s, inj' := ⋯ } a = f) → f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i
Module: Mathlib.Data.Finset.Finsupp

Goal state before:
pp: case refine_2
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι → Finset α
⊢ (f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i) → ∃ a ∈ s.pi t, { toFun := indicator s, inj' := ⋯ } a = f

Tactic:
·

Goal state after:
case refine_2
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι → Finset α
⊢ (f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i) → ∃ a ∈ s.pi t, { toFun := indicator s, inj' := ⋯ } a = f
Module: Mathlib.Data.Finset.Finsupp

Goal state before:
pp: case refine_1
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι →₀ Finset α
ht : t.support ⊆ s
i : ι
h : (i ∈ f.support → i ∈ s) ∧ (i ∈ s → f i ∈ t i)
⊢ f i ∈ t i

pp: case refine_2
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι →₀ Finset α
ht : t.support ⊆ s
i : ι
h : f i ∈ t i
hi : i ∈ f.support
H : t i = 0
⊢ f i = 0

Tactic:
·

Goal state after:
case refine_1
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι →₀ Finset α
ht : t.support ⊆ s
i : ι
h : (i ∈ f.support → i ∈ s) ∧ (i ∈ s → f i ∈ t i)
⊢ f i ∈ t i
Module: Mathlib.Data.Finset.Finsupp

Goal state before:
pp: case pos
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι →₀ Finset α
ht : t.support ⊆ s
i : ι
h : (i ∈ f.support → i ∈ s) ∧ (i ∈ s → f i ∈ t i)
hi : i ∈ s
⊢ f i ∈ t i

pp: case neg
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι →₀ Finset α
ht : t.support ⊆ s
i : ι
h : (i ∈ f.support → i ∈ s) ∧ (i ∈ s → f i ∈ t i)
hi : i ∉ s
⊢ f i ∈ t i

Tactic:
·

Goal state after:
case pos
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι →₀ Finset α
ht : t.support ⊆ s
i : ι
h : (i ∈ f.support → i ∈ s) ∧ (i ∈ s → f i ∈ t i)
hi : i ∈ s
⊢ f i ∈ t i
Module: Mathlib.Data.Finset.Finsupp

Goal state before:
pp: case neg
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι →₀ Finset α
ht : t.support ⊆ s
i : ι
h : (i ∈ f.support → i ∈ s) ∧ (i ∈ s → f i ∈ t i)
hi : i ∉ s
⊢ f i ∈ t i

Tactic:
·

Goal state after:
case neg
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι →₀ Finset α
ht : t.support ⊆ s
i : ι
h : (i ∈ f.support → i ∈ s) ∧ (i ∈ s → f i ∈ t i)
hi : i ∉ s
⊢ f i ∈ t i
Module: Mathlib.Data.Finset.Finsupp

Goal state before:
pp: case refine_2
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι →₀ Finset α
ht : t.support ⊆ s
i : ι
h : f i ∈ t i
hi : i ∈ f.support
H : t i = 0
⊢ f i = 0

Tactic:
·

Goal state after:
case refine_2
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι →₀ Finset α
ht : t.support ⊆ s
i : ι
h : f i ∈ t i
hi : i ∈ f.support
H : t i = 0
⊢ f i = 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case left
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
⊢ ∃ d, (coeff d) (f + g) ≠ 0 ∧ (weight w) d = n

pp: case right
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
⊢ ∀ (d : σ →₀ ℕ), (weight w) d < n → (coeff d) (f + g) = 0

Tactic:
·

Goal state after:
case left
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
⊢ ∃ d, (coeff d) (f + g) ≠ 0 ∧ (weight w) d = n
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case right
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
⊢ ∀ (d : σ →₀ ℕ), (weight w) d < n → (coeff d) (f + g) = 0

Tactic:
·

Goal state after:
case right
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
⊢ ∀ (d : σ →₀ ℕ), (weight w) d < n → (coeff d) (f + g) = 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv : v.intValuation b ≤ v.intValuation a
γ : Multiplicative ℤ
ha : a = 0
⊢ ∃ y, v.intValuation (b - y * a) < ↑γ

pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv : v.intValuation b ≤ v.intValuation a
γ : Multiplicative ℤ
ha : ¬a = 0
⊢ ∃ y, v.intValuation (b - y * a) < ↑γ

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv : v.intValuation b ≤ v.intValuation a
γ : Multiplicative ℤ
ha : a = 0
⊢ ∃ y, v.intValuation (b - y * a) < ↑γ
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
a b : R
hv : v.intValuation b ≤ v.intValuation a
γ : Multiplicative ℤ
ha : ¬a = 0
⊢ ∃ y, v.intValuation (b - y * a) < ↑γ

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
a b : R
hv : v.intValuation b ≤ v.intValuation a
γ : Multiplicative ℤ
ha : ¬a = 0
⊢ ∃ y, v.intValuation (b - y * a) < ↑γ
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case inr
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
this :
  ∀ {σ : Type u_1} {R : Type u_2} [inst : Semiring R] (w : σ → ℕ) {f g : MvPowerSeries σ R},
    weightedOrder w f ≠ weightedOrder w g →
      weightedOrder w f < weightedOrder w g → weightedOrder w (f + g) ≤ min (weightedOrder w f) (weightedOrder w g)
H₁ : ¬weightedOrder w f < weightedOrder w g
⊢ weightedOrder w (f + g) ≤ min (weightedOrder w f) (weightedOrder w g)

pp: σ✝ : Type u_1
R✝ : Type u_2
inst✝¹ : Semiring R✝
w✝ : σ✝ → ℕ
f✝ g✝ : MvPowerSeries σ✝ R✝
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
H₁ : weightedOrder w f < weightedOrder w g
⊢ weightedOrder w (f + g) ≤ min (weightedOrder w f) (weightedOrder w g)

Tactic:
·

Goal state after:
case inr
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
this :
  ∀ {σ : Type u_1} {R : Type u_2} [inst : Semiring R] (w : σ → ℕ) {f g : MvPowerSeries σ R},
    weightedOrder w f ≠ weightedOrder w g →
      weightedOrder w f < weightedOrder w g → weightedOrder w (f + g) ≤ min (weightedOrder w f) (weightedOrder w g)
H₁ : ¬weightedOrder w f < weightedOrder w g
⊢ weightedOrder w (f + g) ≤ min (weightedOrder w f) (weightedOrder w g)
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
S : Type u_3
inst✝¹ : CommSemiring R
s : Finset (σ →₀ ℕ)
inst✝ : CommSemiring S
f : R →+* S
p : MvPowerSeries σ R
x : σ →₀ ℕ
h✝ : x ∈ s
⊢ MvPolynomial.coeff x ((truncFinset S s) ((map f) p)) =
    MvPolynomial.coeff x ((MvPolynomial.map f) ((truncFinset R s) p))

Tactic:
grind [coeff_map, MvPolynomial.coeff_map]

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

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
S : Type u_3
inst✝¹ : CommSemiring R
s : Finset (σ →₀ ℕ)
inst✝ : CommSemiring S
f : R →+* S
p : MvPowerSeries σ R
x : σ →₀ ℕ
h✝ : x ∉ s
⊢ MvPolynomial.coeff x ((truncFinset S s) ((map f) p)) =
    MvPolynomial.coeff x ((MvPolynomial.map f) ((truncFinset R s) p))

Tactic:
grind [coeff_map, MvPolynomial.coeff_map]

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

Goal state before:
pp: case pos
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 : ↑((weight w) i) < weightedOrder w f
⊢ (coeff (i, j).1) f * (coeff (i, j).2) g = 0

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:
·

Goal state after:
case pos
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 : ↑((weight w) i) < weightedOrder w f
⊢ (coeff (i, j).1) f * (coeff (i, j).2) 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:
·

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

Goal state before:
pp: case pos
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)
hj : ↑((weight w) j) < weightedOrder w g
⊢ (coeff (i, j).1) f * (coeff (i, j).2) g = 0

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)
hj : weightedOrder w g ≤ ↑((weight w) j)
⊢ (coeff (i, j).1) f * (coeff (i, j).2) g = 0

Tactic:
·

Goal state after:
case pos
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)
hj : ↑((weight w) j) < weightedOrder w g
⊢ (coeff (i, j).1) f * (coeff (i, j).2) 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)
hj : weightedOrder w g ≤ ↑((weight w) j)
⊢ (coeff (i, j).1) f * (coeff (i, j).2) g = 0

Tactic:
·

Goal state after:
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)
hj : weightedOrder w g ≤ ↑((weight w) j)
⊢ (coeff (i, j).1) f * (coeff (i, j).2) g = 0
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
s : Finset (σ →₀ ℕ)
hs : IsLowerSet ↑s
k : ℕ
p : MvPowerSeries σ R
n : ℕ
hmn : 1 ≤ n
ih : (truncFinset R s) (↑((truncFinset R s) p) ^ n) = (truncFinset R s) (p ^ n)
x : σ →₀ ℕ
hx : x ∈ s
⊢ MvPolynomial.coeff x ((truncFinset R s) (↑((truncFinset R s) p) ^ (n + 1))) =
    MvPolynomial.coeff x ((truncFinset R s) (p ^ (n + 1)))

pp: case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
s : Finset (σ →₀ ℕ)
hs : IsLowerSet ↑s
k : ℕ
p : MvPowerSeries σ R
n : ℕ
hmn : 1 ≤ n
ih : (truncFinset R s) (↑((truncFinset R s) p) ^ n) = (truncFinset R s) (p ^ n)
x : σ →₀ ℕ
hx : x ∉ s
⊢ MvPolynomial.coeff x ((truncFinset R s) (↑((truncFinset R s) p) ^ (n + 1))) =
    MvPolynomial.coeff x ((truncFinset R s) (p ^ (n + 1)))

Tactic:
·

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
s : Finset (σ →₀ ℕ)
hs : IsLowerSet ↑s
k : ℕ
p : MvPowerSeries σ R
n : ℕ
hmn : 1 ≤ n
ih : (truncFinset R s) (↑((truncFinset R s) p) ^ n) = (truncFinset R s) (p ^ n)
x : σ →₀ ℕ
hx : x ∈ s
⊢ MvPolynomial.coeff x ((truncFinset R s) (↑((truncFinset R s) p) ^ (n + 1))) =
    MvPolynomial.coeff x ((truncFinset R s) (p ^ (n + 1)))
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: case refine_1
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
s : Finset (σ →₀ ℕ)
p : MvPolynomial σ R
h : (truncFinset R s) ↑p = p
⊢ p.support ⊆ s

pp: case refine_2
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
s : Finset (σ →₀ ℕ)
p : MvPolynomial σ R
h : p.support ⊆ s
x : σ →₀ ℕ
⊢ MvPolynomial.coeff x ((truncFinset R s) ↑p) = MvPolynomial.coeff x p

Tactic:
·

Goal state after:
case refine_1
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
s : Finset (σ →₀ ℕ)
p : MvPolynomial σ R
h : (truncFinset R s) ↑p = p
⊢ p.support ⊆ s
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case insert
σ : Type u_1
w : σ → ℕ
R : Type u_4
ι : Type u_5
inst✝ : CommRing R
d : σ →₀ ℕ
f : MvPowerSeries σ R
g : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (coeff d) (f * ∏ i ∈ s, (1 - g i)) = (coeff d) f
h : ∀ i ∈ insert a s, ↑((weight w) d) < weightedOrder w (g i)
⊢ (coeff d) (f * ∏ i ∈ insert a s, (1 - g i)) = (coeff d) f

Tactic:
simp only [Finset.mem_insert, forall_eq_or_imp] at h
    rw [Finset.prod_insert ha, ← mul_assoc, mul_right_comm,
      coeff_mul_left_one_sub_of_lt_weightedOrder w h.1, ih h.2]

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

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
x : σ →₀ ℕ
h : degree x < n
⊢ MvPolynomial.coeff x ((truncTotal n) p) = if degree x < n then (coeff x) p else 0

pp: case neg
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
x : σ →₀ ℕ
h : ¬degree x < n
⊢ MvPolynomial.coeff x ((truncTotal n) p) = if degree x < n then (coeff x) p else 0

Tactic:
·

Goal state after:
case pos
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
x : σ →₀ ℕ
h : degree x < n
⊢ MvPolynomial.coeff x ((truncTotal n) p) = if degree x < n then (coeff x) p else 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case refine_1
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
⊢ weightedOrder w (p.toSubring T hp) ≤ weightedOrder w p

pp: case refine_2
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
⊢ weightedOrder w p ≤ weightedOrder w (p.toSubring T hp)

Tactic:
·

Goal state after:
case refine_1
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
⊢ weightedOrder w (p.toSubring T hp) ≤ weightedOrder w p