-- Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic
Goal:
R_1 : Type u_1
inst : CommRing R_1
inst_1 : ValuativeRel R_1
R : Type u_2
Γ : Type u_3
inst_2 : CommRing R
inst_3 : ValuativeRel R
inst_4 : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst_5 : v.Compatible
x_1 x_2 : ValueGroupWithZero R
x y z : R
hz : 0 <ᵥ z
w : R
hw : 0 <ᵥ w
pos :
  Eq.{u_3 + 1} (α := Γ) ((v : (a : R) → Γ) y : Γ) (0 : Γ) →
    Eq.{u_3 + 1} (α := Γ) ((v : (a : R) → Γ) x : Γ) (0 : Γ) →
      0 /
          HMul.hMul (α := ValueGroup₀ v) (β := ValueGroup₀ v)
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) z) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v))
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) w) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v)) =
        HDiv.hDiv (α := ValueGroup₀ v) (β := ValueGroup₀ v) (0 : ValueGroup₀ v)
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) z) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v)) *
          HDiv.hDiv (α := ValueGroup₀ v) (β := ValueGroup₀ v) (0 : ValueGroup₀ v)
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) w) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v))
neg :
  Eq.{u_3 + 1} (α := Γ) ((v : (a : R) → Γ) y : Γ) (0 : Γ) →
    ∀ (h : ¬Eq.{u_3 + 1} (α := Γ) ((v : (a : R) → Γ) x : Γ) (0 : Γ)),
      0 /
          HMul.hMul (α := ValueGroup₀ v) (β := ValueGroup₀ v)
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) z) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v))
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) w) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v)) =
        HDiv.hDiv (α := ValueGroup₀ v) (β := ValueGroup₀ v)
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) x) h) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v))
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) z) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v)) *
          HDiv.hDiv (α := ValueGroup₀ v) (β := ValueGroup₀ v) (0 : ValueGroup₀ v)
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) w) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v))
pos₁ :
  ∀ (h : ¬Eq.{u_3 + 1} (α := Γ) ((v : (a : R) → Γ) y : Γ) (0 : Γ)),
    Eq.{u_3 + 1} (α := Γ) ((v : (a : R) → Γ) x : Γ) (0 : Γ) →
      0 /
          HMul.hMul (α := ValueGroup₀ v) (β := ValueGroup₀ v)
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) z) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v))
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) w) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v)) =
        HDiv.hDiv (α := ValueGroup₀ v) (β := ValueGroup₀ v) (0 : ValueGroup₀ v)
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) z) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v)) *
          HDiv.hDiv (α := ValueGroup₀ v) (β := ValueGroup₀ v)
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) y) h) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v))
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) w) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v))
neg₁ :
  ∀ (h : ¬Eq.{u_3 + 1} (α := Γ) ((v : (a : R) → Γ) y : Γ) (0 : Γ))
    (h_1 : ¬Eq.{u_3 + 1} (α := Γ) ((v : (a : R) → Γ) x : Γ) (0 : Γ)),
    HMul.hMul (α := ValueGroup₀ v) (β := ValueGroup₀ v)
          ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                  (Units.mk0 ((v : (a : R) → Γ) x) ⋯) ⋯) :
              WithZero ↥(valueGroup v)) :
            WithZero ↥(valueGroup v))
          ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                  (Units.mk0 ((v : (a : R) → Γ) y) ⋯) ⋯) :
              WithZero ↥(valueGroup v)) :
            WithZero ↥(valueGroup v)) /
        HMul.hMul (α := ValueGroup₀ v) (β := ValueGroup₀ v)
          ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                  (Units.mk0 ((v : (a : R) → Γ) z) ⋯) ⋯) :
              WithZero ↥(valueGroup v)) :
            WithZero ↥(valueGroup v))
          ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                  (Units.mk0 ((v : (a : R) → Γ) w) ⋯) ⋯) :
              WithZero ↥(valueGroup v)) :
            WithZero ↥(valueGroup v)) =
      HDiv.hDiv (α := ValueGroup₀ v) (β := ValueGroup₀ v)
          ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                  (Units.mk0 ((v : (a : R) → Γ) x) h_1) ⋯) :
              WithZero ↥(valueGroup v)) :
            WithZero ↥(valueGroup v))
          ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                  (Units.mk0 ((v : (a : R) → Γ) z) ⋯) ⋯) :
              WithZero ↥(valueGroup v)) :
            WithZero ↥(valueGroup v)) *
        HDiv.hDiv (α := ValueGroup₀ v) (β := ValueGroup₀ v)
          ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                  (Units.mk0 ((v : (a : R) → Γ) y) h) ⋯) :
              WithZero ↥(valueGroup v)) :
            WithZero ↥(valueGroup v))
          ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                  (Units.mk0 ((v : (a : R) → Γ) w) ⋯) ⋯) :
              WithZero ↥(valueGroup v)) :
            WithZero ↥(valueGroup v))
⊢ (if h : Eq.{u_3 + 1} (α := Γ) ((v : (a : R) → Γ) y : Γ) (0 : Γ) then 0
      else
        if h : Eq.{u_3 + 1} (α := Γ) ((v : (a : R) → Γ) x : Γ) (0 : Γ) then 0
        else
          HMul.hMul (α := ValueGroup₀ v) (β := ValueGroup₀ v)
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) x) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v))
            ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                    (Units.mk0 ((v : (a : R) → Γ) y) ⋯) ⋯) :
                WithZero ↥(valueGroup v)) :
              WithZero ↥(valueGroup v))) /
      HMul.hMul (α := ValueGroup₀ v) (β := ValueGroup₀ v)
        ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                (Units.mk0 ((v : (a : R) → Γ) z) ⋯) ⋯) :
            WithZero ↥(valueGroup v)) :
          WithZero ↥(valueGroup v))
        ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                (Units.mk0 ((v : (a : R) → Γ) w) ⋯) ⋯) :
            WithZero ↥(valueGroup v)) :
          WithZero ↥(valueGroup v)) =
    HDiv.hDiv (α := ValueGroup₀ v) (β := ValueGroup₀ v)
        (if h : Eq.{u_3 + 1} (α := Γ) ((v : (a : R) → Γ) x : Γ) (0 : Γ) then 0
          else
            (↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                  (Units.mk0 ((v : (a : R) → Γ) x) h) ⋯) :
              WithZero ↥(valueGroup v)) :
          ValueGroup₀ v)
        ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                (Units.mk0 ((v : (a : R) → Γ) z) ⋯) ⋯) :
            WithZero ↥(valueGroup v)) :
          WithZero ↥(valueGroup v)) *
      HDiv.hDiv (α := ValueGroup₀ v) (β := ValueGroup₀ v)
        (if h : Eq.{u_3 + 1} (α := Γ) ((v : (a : R) → Γ) y : Γ) (0 : Γ) then 0
          else
            (↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                  (Units.mk0 ((v : (a : R) → Γ) y) h) ⋯) :
              WithZero ↥(valueGroup v)) :
          ValueGroup₀ v)
        ((↑(Subtype.mk (p := fun (x : Γˣ) ↦ Membership.mem (γ := Subgroup Γˣ) (valueGroup v) x)
                (Units.mk0 ((v : (a : R) → Γ) w) ⋯) ⋯) :
            WithZero ↥(valueGroup v)) :
          WithZero ↥(valueGroup v))

Tactic:
split_ifs