Basic Algebra Theories

Associativity

module 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

Commutativity

module Comm

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

end

Associativity and Commutativity

module AC

  clone export Assoc with axiom Assoc
  clone export Comm with type t = t, function op = op, axiom Comm
  meta AC function op

end

Monoids

module Monoid

  clone export Assoc with axiom Assoc
  constant unit : t
  axiom Unit_def_l : forall x:t. op unit x = x
  axiom Unit_def_r : forall x:t. op x unit = x

end

Commutative Monoids

module CommutativeMonoid

  clone export Monoid with axiom Assoc, axiom Unit_def_l, axiom Unit_def_r
  clone export Comm with type t = t, function op = op, axiom Comm
  meta AC function op

end

Groups

module Group

  clone export Monoid with axiom Assoc, axiom Unit_def_l, axiom Unit_def_r
  function inv t : t
  axiom Inv_def_l : forall x:t. op (inv x) x = unit
  axiom Inv_def_r : forall x:t. op x (inv x) = unit

end

Commutative Groups

module CommutativeGroup

  clone export Group with axiom .
  clone export Comm with type t = t, function op = op, axiom Comm
  meta AC function op

end

Rings

module Ring

  type t
  constant zero : t
  function (+) t t : t
  function (-_) t : t
  function (*) t t : t

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

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

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

end

Commutative Rings

module CommutativeRing

  clone export Ring with axiom .
  clone Comm as MulComm with type t = t, function op = (*), axiom Comm
  meta AC function (*)

end

Commutative Rings with Unit

module UnitaryCommutativeRing

  clone export CommutativeRing with axiom .
  constant one : t
  axiom Unitary : forall x:t. one * x = x
  axiom NonTrivialRing : zero <> one

end

Ordered Commutative Rings

module OrderedUnitaryCommutativeRing

  clone export UnitaryCommutativeRing with axiom .

  predicate (<=) t t

  clone export relations.TotalOrder with
    type t = t, predicate rel = (<=), axiom .

  axiom ZeroLessOne : zero <= one

  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
  meta "remove_unused:dependency" axiom CompatOrderMult, function (*)

end

Field theory

module Field

  clone export UnitaryCommutativeRing with axiom .

  function inv t : t

  axiom Inverse : forall x:t. x <> zero -> x * inv x = one

  function (-) (x y : t) : t = x + -y
  function (/) (x y : t) : t = x * inv y

  lemma add_div :
    forall x y z : t. z <> zero -> (x+y)/z = x/z + y/z
  meta "remove_unused:dependency" lemma add_div, function (/)

  lemma sub_div :
    forall x y z : t. z <> zero -> (x-y)/z = x/z - y/z
  meta "remove_unused:dependency" lemma sub_div, function (/)

  lemma neg_div :
    forall x y : t. y <> zero -> (-x)/y = -(x/y)
  meta "remove_unused:dependency" lemma neg_div, function (/)

  lemma assoc_mul_div: forall x y z:t.
    (* todo: discard the hypothesis ? *)
     z <> zero -> (x*y)/z = x*(y/z)
  meta "remove_unused:dependency" lemma assoc_mul_div, function (/)

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

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

end

Ordered Fields

module OrderedField

  clone export Field with axiom .

  predicate (<=) t t

  clone export relations.TotalOrder with
    type t = t, predicate rel = (<=), axiom .

  axiom ZeroLessOne : zero <= one

  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