theorem ex5 : ∀ {a d b c : Nat}, a + d + b > b + 1 + (a + (c + c) + d) → False :=
fun {a d b c} h =>
  False.elim
    (Eq.mp
      (Eq.trans gt_iff_lt._simp_1
        (id
          (Nat.Linear.ExprCnstr.eq_false_of_isUnsat
            (Lean.RArray.branch 2 (Lean.RArray.branch 1 (Lean.RArray.leaf a) (Lean.RArray.leaf d))
              (Lean.RArray.branch 3 (Lean.RArray.leaf b) (Lean.RArray.leaf c)))
            { eq := false,
              lhs :=
                (((Nat.Linear.Expr.var 2).add (Nat.Linear.Expr.num 1)).add
                      (((Nat.Linear.Expr.var 0).add ((Nat.Linear.Expr.var 3).add (Nat.Linear.Expr.var 3))).add
                        (Nat.Linear.Expr.var 1))).add
                  (Nat.Linear.Expr.num 1),
              rhs := ((Nat.Linear.Expr.var 0).add (Nat.Linear.Expr.var 1)).add (Nat.Linear.Expr.var 2) }
            (eagerReduce (Eq.refl true)))))
      h)
theorem ex6 : ∀ {a b : Nat} (p : Nat → Prop), p (a + 1 + a + 2 + b) → p (2 * a + b + 3) :=
fun {a b} p h =>
  Eq.mp
    (congrArg p
      (id
        (Nat.Linear.Expr.eq_of_toNormPoly_eq (Lean.RArray.branch 1 (Lean.RArray.leaf a) (Lean.RArray.leaf b))
          (((((Nat.Linear.Expr.var 0).add (Nat.Linear.Expr.num 1)).add (Nat.Linear.Expr.var 0)).add
                (Nat.Linear.Expr.num 2)).add
            (Nat.Linear.Expr.var 1))
          (((Nat.Linear.Expr.mulL 2 (Nat.Linear.Expr.var 0)).add (Nat.Linear.Expr.var 1)).add (Nat.Linear.Expr.num 3))
          (eagerReduce (Eq.refl true)))))
    h