open Lean.Grind

section IntModule

variable (M : Type) [IntModule M]

example (x y : M) : 2 * x + 3 * y + x = 3 * (x + y) := by grind

variable [LinearOrder M] [OrderedAdd M]

example {x y : M} (h : x ≤ y) : 2 * x + y ≤ 3 * y := by grind

end IntModule

-- We could solve these problems by embedding the NatModule in its Grothendieck completion.
section NatModule

variable (M : Type) [NatModule M]

example (x y : M) : 2 * x + 3 * y + x = 3 * (x + y) := by grind

variable [LinearOrder M] [OrderedAdd M]

example {x y : M} (h : x ≤ y) : 2 * x + y ≤ 3 * y := by grind


end NatModule