Module: Mathlib.Algebra.Polynomial.RuleOfSigns

Goal state before:
pp: case zero.zero
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.zero
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.zero
⊢ SignType.zero = -SignType.zero

pp: case zero.neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.zero
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.neg
⊢ SignType.zero = -SignType.neg

pp: case zero.pos
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.zero
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.pos
⊢ SignType.zero = -SignType.pos

pp: case neg.zero
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.neg
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.zero
⊢ SignType.neg = -SignType.zero

pp: case neg.neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.neg
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.neg
⊢ SignType.neg = -SignType.neg

pp: case neg.pos
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.neg
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.pos
⊢ SignType.neg = -SignType.pos

pp: case pos.zero
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.pos
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.zero
⊢ SignType.pos = -SignType.zero

pp: case pos.neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.pos
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.neg
⊢ SignType.pos = -SignType.neg

pp: case pos.pos
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.pos
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.pos
⊢ SignType.pos = -SignType.pos

Tactic:
<;>

Goal state after:
case zero.zero
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.zero
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.zero
⊢ SignType.zero = -SignType.zero

case zero.neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.zero
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.neg
⊢ SignType.zero = -SignType.neg

case zero.pos
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.zero
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.pos
⊢ SignType.zero = -SignType.pos

case neg.zero
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.neg
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.zero
⊢ SignType.neg = -SignType.zero

case neg.neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.neg
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.neg
⊢ SignType.neg = -SignType.neg

case neg.pos
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.neg
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.pos
⊢ SignType.neg = -SignType.pos

case pos.zero
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.pos
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.zero
⊢ SignType.pos = -SignType.zero

case pos.neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.pos
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.neg
⊢ SignType.pos = -SignType.neg

case pos.pos
R : Type u_1
inst✝¹ : Semiring R
inst✝ : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : SignType.sign P.leadingCoeff ≠ 0
cs : List R
h_eL : P.eraseLead.coeffList = P.eraseLead.leadingCoeff :: cs
h₁ : SignType.sign P.eraseLead.leadingCoeff ≠ 0
h₄ : ¬SignType.sign P.leadingCoeff = SignType.sign P.eraseLead.leadingCoeff
h✝¹ : SignType.sign P.leadingCoeff = SignType.pos
h✝ : SignType.sign P.eraseLead.leadingCoeff = SignType.pos
⊢ SignType.pos = -SignType.pos