-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Semiring R
inst_1 : LinearOrder R
P : R[X]
h : P ≠ 0
hpz : ¬P = 0
hsl : Ne (α := SignType) ((SignType.sign : (a : R) → SignType) P.leadingCoeff : SignType) (0 : SignType)
c : R
cs : List R
h_eL : P.eraseLead.coeffList = c :: cs
neg :
  List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2)
          (match (motive := Bool → List SignType) decide ¬False with
          | true =>
            (SignType.sign : (a : R) → SignType) P.leadingCoeff ::
              HAppend.hAppend (α := List SignType) (β := List SignType)
                (if
                      (decide ¬Eq (α := SignType) ((SignType.sign : (a : R) → SignType) 0 : SignType) (0 : SignType)) =
                        true then
                    List.replicate (P.natDegree - P.eraseLead.degree.succ) ((SignType.sign : (a : R) → SignType) 0)
                  else [] :
                  List SignType)
                (match (motive := Bool → List SignType)
                    decide ¬Eq (α := SignType) ((SignType.sign : (a : R) → SignType) c : SignType) (0 : SignType) with
                  | true =>
                    (SignType.sign : (a : R) → SignType) c ::
                      List.filter (fun (x : SignType) ↦ decide ¬x = 0)
                        (List.map (⇑SignType.sign : (a : R) → SignType) cs)
                  | false =>
                    List.filter (fun (x : SignType) ↦ decide ¬x = 0)
                      (List.map (⇑SignType.sign : (a : R) → SignType) cs) :
                  List SignType)
          | false =>
            HAppend.hAppend (α := List SignType) (β := List SignType)
              (if
                    (decide ¬Eq (α := SignType) ((SignType.sign : (a : R) → SignType) 0 : SignType) (0 : SignType)) =
                      true then
                  List.replicate (P.natDegree - P.eraseLead.degree.succ) ((SignType.sign : (a : R) → SignType) 0)
                else [] :
                List SignType)
              (match (motive := Bool → List SignType)
                  decide ¬Eq (α := SignType) ((SignType.sign : (a : R) → SignType) c : SignType) (0 : SignType) with
                | true =>
                  (SignType.sign : (a : R) → SignType) c ::
                    List.filter (fun (x : SignType) ↦ decide ¬x = 0) (List.map (⇑SignType.sign : (a : R) → SignType) cs)
                | false =>
                  List.filter (fun (x : SignType) ↦ decide ¬x = 0) (List.map (⇑SignType.sign : (a : R) → SignType) cs) :
                List SignType))) -
      1 =
    List.length (α := SignType)
          (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2)
            (match (motive := Bool → List SignType)
              decide ¬Eq (α := SignType) ((SignType.sign : (a : R) → SignType) c : SignType) (0 : SignType) with
            | true =>
              (SignType.sign : (a : R) → SignType) c ::
                List.filter (fun (x : SignType) ↦ decide ¬x = 0) (List.map (⇑SignType.sign : (a : R) → SignType) cs)
            | false =>
              List.filter (fun (x : SignType) ↦ decide ¬x = 0) (List.map (⇑SignType.sign : (a : R) → SignType) cs))) -
        1 +
      if
          Eq (α := SignType) ((SignType.sign : (a : R) → SignType) P.leadingCoeff : SignType)
            (-(SignType.sign : (a : R) → SignType) P.eraseLead.leadingCoeff : SignType) then
        1
      else 0
⊢ List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ x1 ≠ x2)
            (List.filter (fun (x : SignType) ↦ decide (x ≠ 0))
              ((SignType.sign : (a : R) → SignType) P.leadingCoeff ::
                HAppend.hAppend (α := List SignType) (β := List SignType)
                  (List.replicate (P.natDegree - P.eraseLead.degree.succ) ((SignType.sign : (a : R) → SignType) 0) :
                    List SignType)
                  (List.map (⇑SignType.sign : (a : R) → SignType) (c :: cs) : List SignType))) :
          List SignType) -
      1 =
    List.length (α := SignType)
          (List.destutter (fun (x1 x2 : SignType) ↦ x1 ≠ x2)
            (List.filter (fun (x : SignType) ↦ decide (x ≠ 0))
              (List.map (⇑SignType.sign : (a : R) → SignType) (c :: cs)))) -
        1 +
      if
          Eq (α := SignType) ((SignType.sign : (a : R) → SignType) P.leadingCoeff : SignType)
            (-(SignType.sign : (a : R) → SignType) P.eraseLead.leadingCoeff : SignType) then
        1
      else 0

Tactic:
simp only [List.filter_append, List.filter_replicate, List.map_cons, List.filter, ne_eq, hsl]

Premises used: List.map_cons, ne_eq, List.filter, List.filter_append, List.filter_replicate