algebra.why

theory Assoc
  type t
  function op t t : t
  axiom Assoc : forall x y z : t. op (op x y) z = op x (op y z)
end

theory Comm
  type t
  function op t t : t
  axiom Comm : forall x y : t. op x y = op y x
end

theory AC
  type t
  function op t t : t
  clone Assoc with type t = t, function op = op
  clone Comm  with type t = t, function op = op
  meta AC function op (*, prop Assoc.Assoc, prop Comm.Comm*)
end

theory Group
  type t

  function unit : t

  function op t t : t
  function inv t : t

  axiom Unit_def : forall x:t. op x unit = x

  clone Assoc with type t = t, function op = op

  axiom Inv_def : forall x:t. op x (inv x) = unit

  lemma Inv_unit : forall x y:t. op x (inv y) = unit -> x = y

end

theory CommutativeGroup
  clone export Group
  clone Comm with type t = t, function op = op
  meta AC function op (*, prop Assoc.Assoc, prop Comm.Comm*)
end

theory Ring
  type t
  function zero : t
  function (+) t t : t
  function (-_) t : t
  function (*) t t : t

  clone CommutativeGroup with type t = t,
                              function unit = zero,
                              function op = (+),
                              function inv = (-_)

  clone Assoc with type t = t, function op = (*)

  axiom Mul_distr: forall x y z : t. x * (y + z) = x * y + x * z

  function (-) (x y : t) : t = x + -y
end

theory CommutativeRing
  clone export Ring
  clone Comm with type t = t, function op = (*)
  meta AC function (*) (*, prop Assoc.Assoc, prop Comm.Comm*)
end

theory UnitaryCommutativeRing
  clone export CommutativeRing
  function one : t
  axiom Unitary : forall x:t. one * x = x
  axiom NonTrivialRing : zero <> one
end

theory OrderedUnitaryCommutativeRing
  clone export UnitaryCommutativeRing
  predicate (<=) t t
  predicate (>=) (x y : t) = y <= x
  clone export relations.TotalOrder with type t = t, predicate rel = (<=)

  axiom CompatOrderAdd  :
    forall x y z : t. x <= y -> x + z <= y + z
  axiom CompatOrderMult :
    forall x y z : t. x <= y -> zero <= z -> x * z <= y * z
end

theory Field
  clone export UnitaryCommutativeRing
  function inv t : t
  axiom Inverse : forall x:t. x <> zero -> x * inv x = one
  function (/) (x y : t) : t = x * inv y

  lemma assoc_mul_div: forall x y z:t.
    (* todo: discard the hypothesis ? *)
     z <> zero -> (x*y)/z = x*(y/z)

  lemma assoc_div_mul: forall x y z:t.
    (* todo: discard the hypothesis ? *)
     y <> zero /\ z <> zero -> (x/y)/z = x/(y*z)

  lemma assoc_div_div: forall x y z:t.
    (* todo: discard the hypothesis ? *)
     y <> zero /\ z <> zero -> x/(y/z) = (x*z)/y

end

theory OrderedField
  clone export Field

  predicate (<=) t t
  predicate (>=) (x y : t) = y <= x
  clone export relations.TotalOrder with type t = t, predicate rel = (<=)

  axiom CompatOrderAdd  : forall x y z : t. x <= y -> x + z <= y + z
  axiom CompatOrderMult :
    forall x y z : t. x <= y -> zero <= z -> x * z <= y * z
end

(* to be discussed: should we add the following lemmas, and where

  lemma InvMult : forall x y : t. (-x) * y = - (x * y) = x * (-y)
  lemma InvSquare : forall x : t. x * x = (-x) * (-x)
  lemma ZeroMult : forall x : t. x * zero = zero = zero * x
  lemma SquareNonNeg1 : forall x : t. x <= zero -> zero <= x * x
  lemma SquareNonNeg : forall x : t. zero <= x * x
  lemma ZeroLessOne : zero <= one
*)