mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities

Goal state before:
pp: case neg
σ : Type u_1
inst✝ : DecidableEq σ
t : Finset σ × σ
h1 : t.2 ∈ t.1
h2 : (t.1.erase t.2, t.2).2 ∉ (t.1.erase t.2, t.2).1
⊢ (cons (t.1.erase t.2, t.2).2 (t.1.erase t.2, t.2).1 ⋯, (t.1.erase t.2, t.2).2) = t

pp: case pos
σ : Type u_1
inst✝ : DecidableEq σ
t : Finset σ × σ
h1 : t.2 ∉ t.1
h3 : (cons t.2 t.1 h1, t.2).2 ∈ (cons t.2 t.1 h1, t.2).1
⊢ ((cons t.2 t.1 h1, t.2).1.erase (cons t.2 t.1 h1, t.2).2, (cons t.2 t.1 h1, t.2).2) = t

pp: case neg
σ : Type u_1
inst✝ : DecidableEq σ
t : Finset σ × σ
h1 : t.2 ∉ t.1
h3 : (cons t.2 t.1 h1, t.2).2 ∉ (cons t.2 t.1 h1, t.2).1
⊢ (cons (cons t.2 t.1 h1, t.2).2 (cons t.2 t.1 h1, t.2).1 ⋯, (cons t.2 t.1 h1, t.2).2) = t

Tactic:
·

Goal state after:
case neg
σ : Type u_1
inst✝ : DecidableEq σ
t : Finset σ × σ
h1 : t.2 ∈ t.1
h2 : (t.1.erase t.2, t.2).2 ∉ (t.1.erase t.2, t.2).1
⊢ (cons (t.1.erase t.2, t.2).2 (t.1.erase t.2, t.2).1 ⋯, (t.1.erase t.2, t.2).2) = t
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case h.refine_1
K : Type u_2
inst✝ : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
T : Set (K⸨X⸩ × K⸨X⸩)
pre_T : (fun x ↦ ((algebraMap K⟮X⟯ K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩) x.2)) ⁻¹' T ⊆ S
R : Set K⸨X⸩
hR : R ∈ nhds 0
pre_R : (fun x ↦ x.2 - x.1) ⁻¹' R ⊆ T
d : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
hd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ R
⊢ ∃ γ, {y | Valued.v.restrict y < ↑γ} ⊆ {P | Valued.v P < embedding ↑d}

pp: case h.refine_2
K : Type u_2
inst✝ : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
T : Set (K⸨X⸩ × K⸨X⸩)
pre_T : (fun x ↦ ((algebraMap K⟮X⟯ K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩) x.2)) ⁻¹' T ⊆ S
R : Set K⸨X⸩
hR : R ∈ nhds 0
pre_R : (fun x ↦ x.2 - x.1) ⁻¹' R ⊆ T
d : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
hd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ R
x✝¹ : K⟮X⟯ × K⟮X⟯
x✝ : x✝¹ ∈ (fun x ↦ x.2 - x.1) ⁻¹' {P | Valued.v P < embedding ↑d}
⊢ ((algebraMap K⟮X⟯ K⸨X⸩) x✝¹.1, (algebraMap K⟮X⟯ K⸨X⸩) x✝¹.2) ∈ (fun x ↦ x.2 - x.1) ⁻¹' R

Tactic:
·

Goal state after:
case h.refine_1
K : Type u_2
inst✝ : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
T : Set (K⸨X⸩ × K⸨X⸩)
pre_T : (fun x ↦ ((algebraMap K⟮X⟯ K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩) x.2)) ⁻¹' T ⊆ S
R : Set K⸨X⸩
hR : R ∈ nhds 0
pre_R : (fun x ↦ x.2 - x.1) ⁻¹' R ⊆ T
d : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
hd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ R
⊢ ∃ γ, {y | Valued.v.restrict y < ↑γ} ⊆ {P | Valued.v P < embedding ↑d}
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case inr.refine_2
σ : Type u_1
R : Type u_3
inst✝¹ : CommSemiring R
inst✝ : LinearOrder σ
p : MvPolynomial σ R
hp : p.IsSymmetric
h0 : p ≠ 0
i j : σ
hle : i ≤ j
hlt : (ofLex (supDegree (⇑toLex) p)) i < (ofLex (supDegree (⇑toLex) p)) j
⊢ (ofLex (p.support.sup ⇑toLex)) i < (Finsupp.mapDomain (⇑(Equiv.swap i j)) (ofLex (supDegree (⇑toLex) p))) i

Tactic:
·

Goal state after:
case inr.refine_2
σ : Type u_1
R : Type u_3
inst✝¹ : CommSemiring R
inst✝ : LinearOrder σ
p : MvPolynomial σ R
hp : p.IsSymmetric
h0 : p ≠ 0
i j : σ
hle : i ≤ j
hlt : (ofLex (supDegree (⇑toLex) p)) i < (ofLex (supDegree (⇑toLex) p)) j
⊢ (ofLex (p.support.sup ⇑toLex)) i < (Finsupp.mapDomain (⇑(Equiv.swap i j)) (ofLex (supDegree (⇑toLex) p))) i
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities

Goal state before:
pp: case pos
σ : Type u_1
inst✝ : DecidableEq σ
t : Finset σ × σ
h1 : t.2 ∉ t.1
h3 : (cons t.2 t.1 h1, t.2).2 ∈ (cons t.2 t.1 h1, t.2).1
⊢ ((cons t.2 t.1 h1, t.2).1.erase (cons t.2 t.1 h1, t.2).2, (cons t.2 t.1 h1, t.2).2) = t

pp: case neg
σ : Type u_1
inst✝ : DecidableEq σ
t : Finset σ × σ
h1 : t.2 ∉ t.1
h3 : (cons t.2 t.1 h1, t.2).2 ∉ (cons t.2 t.1 h1, t.2).1
⊢ (cons (cons t.2 t.1 h1, t.2).2 (cons t.2 t.1 h1, t.2).1 ⋯, (cons t.2 t.1 h1, t.2).2) = t

Tactic:
·

Goal state after:
case pos
σ : Type u_1
inst✝ : DecidableEq σ
t : Finset σ × σ
h1 : t.2 ∉ t.1
h3 : (cons t.2 t.1 h1, t.2).2 ∈ (cons t.2 t.1 h1, t.2).1
⊢ ((cons t.2 t.1 h1, t.2).1.erase (cons t.2 t.1 h1, t.2).2, (cons t.2 t.1 h1, t.2).2) = t
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities

Goal state before:
pp: case neg
σ : Type u_1
inst✝ : DecidableEq σ
t : Finset σ × σ
h1 : t.2 ∉ t.1
h3 : (cons t.2 t.1 h1, t.2).2 ∉ (cons t.2 t.1 h1, t.2).1
⊢ (cons (cons t.2 t.1 h1, t.2).2 (cons t.2 t.1 h1, t.2).1 ⋯, (cons t.2 t.1 h1, t.2).2) = t

Tactic:
·

Goal state after:
case neg
σ : Type u_1
inst✝ : DecidableEq σ
t : Finset σ × σ
h1 : t.2 ∉ t.1
h3 : (cons t.2 t.1 h1, t.2).2 ∉ (cons t.2 t.1 h1, t.2).1
⊢ (cons (cons t.2 t.1 h1, t.2).2 (cons t.2 t.1 h1, t.2).1 ⋯, (cons t.2 t.1 h1, t.2).2) = t
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities

Goal state before:
pp: case inl
σ : Type u_1
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h1 : t.2 ∈ t.1
⊢ #(pairMap σ t).1 ≤ k ∧ (#(pairMap σ t).1 = k → (pairMap σ t).2 ∈ (pairMap σ t).1)

pp: case inr
σ : Type u_1
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h1 : t.2 ∉ t.1
⊢ #(pairMap σ t).1 ≤ k ∧ (#(pairMap σ t).1 = k → (pairMap σ t).2 ∈ (pairMap σ t).1)

Tactic:
·

Goal state after:
case inl
σ : Type u_1
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h1 : t.2 ∈ t.1
⊢ #(pairMap σ t).1 ≤ k ∧ (#(pairMap σ t).1 = k → (pairMap σ t).2 ∈ (pairMap σ t).1)
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case refine_1
R : Type u_3
n m : ℕ
inst✝ : CommRing R
h : n ≤ m
p : MvPolynomial (Fin n) R
hp : ¬p = 0
t : Fin n →₀ ℕ
ht : t ∈ p.support
⊢ ↑((esymmAlgHom (Fin m) R n) ((monomial t) (coeff t p))) ≠ 0

pp: case refine_2
R : Type u_3
n m : ℕ
inst✝ : CommRing R
h : n ≤ m
p : MvPolynomial (Fin n) R
hp : ¬p = 0
t : Fin n →₀ ℕ
ht : t ∈ ↑p.support
s : Fin n →₀ ℕ
hs : s ∈ ↑p.support
he :
  (supDegree ⇑toLex ∘ fun i ↦ ↑((esymmAlgHom (Fin m) R n) ((monomial i) (coeff i p)))) t =
    (supDegree ⇑toLex ∘ fun i ↦ ↑((esymmAlgHom (Fin m) R n) ((monomial i) (coeff i p)))) s
⊢ (accumulate n m) ⇑t = (accumulate n m) ⇑s

Tactic:
·

Goal state after:
case refine_1
R : Type u_3
n m : ℕ
inst✝ : CommRing R
h : n ≤ m
p : MvPolynomial (Fin n) R
hp : ¬p = 0
t : Fin n →₀ ℕ
ht : t ∈ p.support
⊢ ↑((esymmAlgHom (Fin m) R n) ((monomial t) (coeff t p))) ≠ 0
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities

Goal state before:
pp: case inr
σ : Type u_1
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h1 : t.2 ∉ t.1
⊢ #(pairMap σ t).1 ≤ k ∧ (#(pairMap σ t).1 = k → (pairMap σ t).2 ∈ (pairMap σ t).1)

Tactic:
·

Goal state after:
case inr
σ : Type u_1
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h1 : t.2 ∉ t.1
⊢ #(pairMap σ t).1 ≤ k ∧ (#(pairMap σ t).1 = k → (pairMap σ t).2 ∈ (pairMap σ t).1)
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case right.inl
R : Type u_3
inst✝ : CommRing R
n : ℕ
hp : 0 ∈ symmetricSubalgebra (Fin n) R
⊢ ⟨0, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range

pp: case right.inr
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
⊢ ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range

Tactic:
·

Goal state after:
case right.inl
R : Type u_3
inst✝ : CommRing R
n : ℕ
hp : 0 ∈ symmetricSubalgebra (Fin n) R
⊢ ⟨0, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities

Goal state before:
pp: case inl
σ : Type u_1
R : Type u_2
inst✝² : CommRing R
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∈ t.1
⊢ (-1) ^ #t.1 * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - #t.1)) +
      (-1) ^ #(pairMap σ t).1 * ((∏ a ∈ (pairMap σ t).1, X a) * X (pairMap σ t).2 ^ (k - #(pairMap σ t).1)) =
    0

pp: case inr
σ : Type u_1
R : Type u_2
inst✝² : CommRing R
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∉ t.1
⊢ (-1) ^ #t.1 * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - #t.1)) +
      (-1) ^ #(pairMap σ t).1 * ((∏ a ∈ (pairMap σ t).1, X a) * X (pairMap σ t).2 ^ (k - #(pairMap σ t).1)) =
    0

Tactic:
·

Goal state after:
case inl
σ : Type u_1
R : Type u_2
inst✝² : CommRing R
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∈ t.1
⊢ (-1) ^ #t.1 * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - #t.1)) +
      (-1) ^ #(pairMap σ t).1 * ((∏ a ∈ (pairMap σ t).1, X a) * X (pairMap σ t).2 ^ (k - #(pairMap σ t).1)) =
    0
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case mpr
K : Type u_2
inst✝ : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
⊢ (∃ t ∈ nhds 0, (fun x ↦ x.2 - x.1) ⁻¹' t ⊆ S) →
    ∃ t,
      (∃ t_1 ∈ nhds 0, (fun x ↦ x.2 - x.1) ⁻¹' t_1 ⊆ t) ∧
        (fun x ↦ ((algebraMap K⟮X⟯ K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩) x.2)) ⁻¹' t ⊆ S

Tactic:
·

Goal state after:
case mpr
K : Type u_2
inst✝ : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
⊢ (∃ t ∈ nhds 0, (fun x ↦ x.2 - x.1) ⁻¹' t ⊆ S) →
    ∃ t,
      (∃ t_1 ∈ nhds 0, (fun x ↦ x.2 - x.1) ⁻¹' t_1 ⊆ t) ∧
        (fun x ↦ ((algebraMap K⟮X⟯ K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩) x.2)) ⁻¹' t ⊆ S
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
⊢ (accumulate n n) ⇑t = ⇑(ofLex (supDegree (⇑toLex) p))

pp: R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
⊢ leadingCoeff (⇑toLex) p ≠ 0

Tactic:
·

Goal state after:
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
⊢ (accumulate n n) ⇑t = ⇑(ofLex (supDegree (⇑toLex) p))
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
⊢ leadingCoeff (⇑toLex) p ≠ 0

Tactic:
·

Goal state after:
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
⊢ leadingCoeff (⇑toLex) p ≠ 0
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case mpr.refine_1
K : Type u_2
inst✝ : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w✝ : Set K⟮X⟯
hT : w✝ ∈ nhds 0
pre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S
d : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
hd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝
X : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}
X_def : X = {f | Valued.v f < embedding ↑d}
⊢ X ∈ nhds 0 ∧ (fun x ↦ x.2 - x.1) ⁻¹' X ⊆ (fun x ↦ x.2 - x.1) ⁻¹' X

pp: case mpr.refine_2
K : Type u_2
inst✝ : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w✝ : Set K⟮X⟯
hT : w✝ ∈ nhds 0
pre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S
d : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
hd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝
X : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}
X_def : X = {f | Valued.v f < embedding ↑d}
⊢ (fun x ↦ ((algebraMap K⟮X⟯ K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩) x.2)) ⁻¹' (fun x ↦ x.2 - x.1) ⁻¹' X ⊆ S

Tactic:
·

Goal state after:
case mpr.refine_1
K : Type u_2
inst✝ : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w✝ : Set K⟮X⟯
hT : w✝ ∈ nhds 0
pre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S
d : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
hd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝
X : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}
X_def : X = {f | Valued.v f < embedding ↑d}
⊢ X ∈ nhds 0 ∧ (fun x ↦ x.2 - x.1) ⁻¹' X ⊆ (fun x ↦ x.2 - x.1) ⁻¹' X
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case mpr.refine_1
K : Type u_2
inst✝ : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w✝ : Set K⟮X⟯
hT : w✝ ∈ nhds 0
pre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S
d : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
hd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝
X : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}
X_def : X = {f | Valued.v f < embedding ↑d}
⊢ X ∈ nhds 0

Tactic:
·

Goal state after:
case mpr.refine_1
K : Type u_2
inst✝ : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w✝ : Set K⟮X⟯
hT : w✝ ∈ nhds 0
pre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S
d : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
hd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝
X : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}
X_def : X = {f | Valued.v f < embedding ↑d}
⊢ X ∈ nhds 0
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case right.inr.ind.inl
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
hd : supDegree (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) = supDegree (⇑toLex) p
he : p = esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)
⊢ ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range

pp: case right.inr.ind.inr
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
hd : supDegree (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) = supDegree (⇑toLex) p
hne : p ≠ esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)
⊢ ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range

Tactic:
·

Goal state after:
case right.inr.ind.inl
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
hd : supDegree (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) = supDegree (⇑toLex) p
he : p = esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)
⊢ ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities

Goal state before:
pp: case inr
σ : Type u_1
R : Type u_2
inst✝² : CommRing R
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∉ t.1
⊢ (-1) ^ #t.1 * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - #t.1)) +
      (-1) ^ #(pairMap σ t).1 * ((∏ a ∈ (pairMap σ t).1, X a) * X (pairMap σ t).2 ^ (k - #(pairMap σ t).1)) =
    0

Tactic:
·

Goal state after:
case inr
σ : Type u_1
R : Type u_2
inst✝² : CommRing R
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
k : ℕ
t : Finset σ × σ
h : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)
h2 : ∀ (n : ℕ), -(-1) ^ n = (-1) ^ (n + 1)
h1 : t.2 ∉ t.1
⊢ (-1) ^ #t.1 * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - #t.1)) +
      (-1) ^ #(pairMap σ t).1 * ((∏ a ∈ (pairMap σ t).1, X a) * X (pairMap σ t).2 ^ (k - #(pairMap σ t).1)) =
    0
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case right.inr.ind.inr.refine_2
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
hd : supDegree (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) = supDegree (⇑toLex) p
hne : p ≠ esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)
this : supDegree (⇑toLex) (p - esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) < supDegree (⇑toLex) p
⊢ ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range

pp: case right.inr.ind.inr.refine_1
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
hd : supDegree (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) = supDegree (⇑toLex) p
hne : p ≠ esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)
⊢ leadingCoeff (⇑toLex) p = leadingCoeff (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p))

Tactic:
·

Goal state after:
case right.inr.ind.inr.refine_2
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
hd : supDegree (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) = supDegree (⇑toLex) p
hne : p ≠ esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)
this : supDegree (⇑toLex) (p - esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) < supDegree (⇑toLex) p
⊢ ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case right.inr.ind.inr.refine_2
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
hd : supDegree (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) = supDegree (⇑toLex) p
hne : p ≠ esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)
this : supDegree (⇑toLex) (p - esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) < supDegree (⇑toLex) p
⊢ p - esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p) ≠ 0

pp: case right.inr.ind.inr.refine_2
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
ih : ⟨p - esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p), ⋯⟩ ∈ (esymmAlgHom (Fin n) R n).range
hd : supDegree (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) = supDegree (⇑toLex) p
hne : p ≠ esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)
this : supDegree (⇑toLex) (p - esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) < supDegree (⇑toLex) p
⊢ ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range

Tactic:
·

Goal state after:
case right.inr.ind.inr.refine_2
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
hd : supDegree (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) = supDegree (⇑toLex) p
hne : p ≠ esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)
this : supDegree (⇑toLex) (p - esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) < supDegree (⇑toLex) p
⊢ p - esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p) ≠ 0
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case mpr.refine_2
K : Type u_2
inst✝ : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w✝ : Set K⟮X⟯
hT : w✝ ∈ nhds 0
pre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S
d : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
hd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝
X : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}
X_def : X = {f | Valued.v f < embedding ↑d}
⊢ (fun x ↦ ((algebraMap K⟮X⟯ K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩) x.2)) ⁻¹' (fun x ↦ x.2 - x.1) ⁻¹' X ⊆ S

Tactic:
·

Goal state after:
case mpr.refine_2
K : Type u_2
inst✝ : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w✝ : Set K⟮X⟯
hT : w✝ ∈ nhds 0
pre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S
d : (MonoidWithZeroHom.ofClass Valued.v).ValueGroup₀ˣ
hd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝
X : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}
X_def : X = {f | Valued.v f < embedding ↑d}
⊢ (fun x ↦ ((algebraMap K⟮X⟯ K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩) x.2)) ⁻¹' (fun x ↦ x.2 - x.1) ⁻¹' X ⊆ S
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case right.inr.ind.inr.refine_1
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
hd : supDegree (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) = supDegree (⇑toLex) p
hne : p ≠ esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)
⊢ leadingCoeff (⇑toLex) p = leadingCoeff (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p))

Tactic:
·

Goal state after:
case right.inr.ind.inr.refine_1
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
hd : supDegree (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)) = supDegree (⇑toLex) p
hne : p ≠ esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p)
⊢ leadingCoeff (⇑toLex) p = leadingCoeff (⇑toLex) (esymmAlgHomMonomial (Fin n) t (leadingCoeff (⇑toLex) p))