Module: Mathlib.Algebra.Polynomial.RuleOfSigns

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
η : R
hη : 0 < η
P : R[X]
hP : P ≠ 0
h_lC : 0 < P.leadingCoeff
h_mul : (X - C η) * P ≠ 0
h_deg_mul : ((X - C η) * P).natDegree = P.natDegree + 1
d : ℕ
ih : ∀ m < d + 1, ∀ {P : R[X]}, P ≠ 0 → P.natDegree = m → P.signVariations + 1 ≤ ((X - C η) * P).signVariations
hd : P.natDegree = d + 1
h_ih : P.eraseLead.natDegree < d + 1
h_mul_lC : SignType.sign ((X - C η) * P).leadingCoeff = 1
h_ηP : 0 < η * P.coeff (d + 1)
hs_nC : SignType.sign P.nextCoeff = 0
hs_nC_mul : SignType.sign ((X - C η) * P).nextCoeff = -1
h✝ : ¬P.eraseLead = 0
this✝ : ((X - C η) * P).eraseLead.signVariations + 1 = ((X - C η) * P).signVariations
this : ((X - C η) * P).eraseLead.eraseLead.signVariations ≤ ((X - C η) * P).eraseLead.signVariations
⊢ ((X - C η) * P).eraseLead.eraseLead.signVariations ≤ ((X - C η) * P).eraseLead.signVariations

pp: R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
η : R
hη : 0 < η
P : R[X]
hP : P ≠ 0
h_lC : 0 < P.leadingCoeff
h_mul : (X - C η) * P ≠ 0
h_deg_mul : ((X - C η) * P).natDegree = P.natDegree + 1
d : ℕ
ih : ∀ m < d + 1, ∀ {P : R[X]}, P ≠ 0 → P.natDegree = m → P.signVariations + 1 ≤ ((X - C η) * P).signVariations
hd : P.natDegree = d + 1
h_ih : P.eraseLead.natDegree < d + 1
h_mul_lC : SignType.sign ((X - C η) * P).leadingCoeff = 1
h_ηP : 0 < η * P.coeff (d + 1)
hs_nC : SignType.sign P.nextCoeff = 0
hs_nC_mul : SignType.sign ((X - C η) * P).nextCoeff = -1
h✝ : ¬P.eraseLead = 0
this✝ : ((X - C η) * P).eraseLead.signVariations + 1 = ((X - C η) * P).signVariations
this : ((X - C η) * P).eraseLead.eraseLead.signVariations ≤ ((X - C η) * P).eraseLead.signVariations
⊢ P.nextCoeff = 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
η : R
hη : 0 < η
P : R[X]
hP : P ≠ 0
h_lC : 0 < P.leadingCoeff
h_mul : (X - C η) * P ≠ 0
h_deg_mul : ((X - C η) * P).natDegree = P.natDegree + 1
d : ℕ
ih : ∀ m < d + 1, ∀ {P : R[X]}, P ≠ 0 → P.natDegree = m → P.signVariations + 1 ≤ ((X - C η) * P).signVariations
hd : P.natDegree = d + 1
h_ih : P.eraseLead.natDegree < d + 1
h_mul_lC : SignType.sign ((X - C η) * P).leadingCoeff = 1
h_ηP : 0 < η * P.coeff (d + 1)
hs_nC : SignType.sign P.nextCoeff = 0
hs_nC_mul : SignType.sign ((X - C η) * P).nextCoeff = -1
h✝ : ¬P.eraseLead = 0
this✝ : ((X - C η) * P).eraseLead.signVariations + 1 = ((X - C η) * P).signVariations
this : ((X - C η) * P).eraseLead.eraseLead.signVariations ≤ ((X - C η) * P).eraseLead.signVariations
⊢ ((X - C η) * P).eraseLead.eraseLead.signVariations ≤ ((X - C η) * P).eraseLead.signVariations

R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
η : R
hη : 0 < η
P : R[X]
hP : P ≠ 0
h_lC : 0 < P.leadingCoeff
h_mul : (X - C η) * P ≠ 0
h_deg_mul : ((X - C η) * P).natDegree = P.natDegree + 1
d : ℕ
ih : ∀ m < d + 1, ∀ {P : R[X]}, P ≠ 0 → P.natDegree = m → P.signVariations + 1 ≤ ((X - C η) * P).signVariations
hd : P.natDegree = d + 1
h_ih : P.eraseLead.natDegree < d + 1
h_mul_lC : SignType.sign ((X - C η) * P).leadingCoeff = 1
h_ηP : 0 < η * P.coeff (d + 1)
hs_nC : SignType.sign P.nextCoeff = 0
hs_nC_mul : SignType.sign ((X - C η) * P).nextCoeff = -1
h✝ : ¬P.eraseLead = 0
this✝ : ((X - C η) * P).eraseLead.signVariations + 1 = ((X - C η) * P).signVariations
this : ((X - C η) * P).eraseLead.eraseLead.signVariations ≤ ((X - C η) * P).eraseLead.signVariations
⊢ P.nextCoeff = 0