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(** {1 Basic Algebra Theories} *)
(** {2 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
(** {2 Commutativity} *)
module Comm
type t
function op t t : t
axiom Comm : forall x y : t. op x y = op y x
end
(** {2 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
(** {2 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
(** {2 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
(** {2 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
(***
lemma Inv_unit : forall x y:t. op x (inv y) = unit -> x = y
*)
end
(** {2 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
(** {2 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
(** {2 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
(** {2 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
(** {2 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
(** {2 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
(** {2 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
(***
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
*)