perga/examples/algebra.pg

-- --------------------------------------------------------------------------------------------------------------
-- |                                            BASIC LOGIC                                                     |
-- --------------------------------------------------------------------------------------------------------------

-- first some basic logic, see <examples/logic.pg> for more details on these definitions

false : * := forall (A : *), A;
false_elim (A : *) (contra : false) : A := contra A;

not (A : *) : * := A -> false;
not_intro (A : *) (h : A -> false) : not A := h;
not_elim (A B : *) (a : A) (na : not A) : B := na a B;

and (A B : *) : * := forall (C : *), (A -> B -> C) -> C;
and_intro (A B : *) (a : A) (b : B) : and A B :=
    fun (C : *) (H : A -> B -> C) => H a b;
and_elim_l (A B : *) (ab : and A B) : A :=
    ab A (fun (a : A) (b : B) => a);
and_elim_r (A B : *) (ab : and A B) : B :=
    ab B (fun (a : A) (b : B) => b);

or (A B : *) : * := forall (C : *), (A -> C) -> (B -> C) -> C;
or_intro_l (A B : *) (a : A) : or A B :=
    fun (C : *) (ha : A -> C) (hb : B -> C) => ha a;
or_intro_r (A B : *) (b : B) : or A B :=
    fun (C : *) (ha : A -> C) (hb : B -> C) => hb b;
or_elim (A B C : *) (ab : or A B) (ha : A -> C) (hb : B -> C) : C :=
    ab C ha hb;

eq (A : *) (x y : A) := forall (P : A -> *), P x -> P y;
eq_refl (A : *) (x : A) : eq A x x := fun (P : A -> *) (Hx : P x) => Hx;
eq_sym (A : *) (x y : A) (Hxy : eq A x y) : eq A y x := fun (P : A -> *) (Hy : P y) =>
    Hxy (fun (z : A) => P z -> P x) (fun (Hx : P x) => Hx) Hy;
eq_trans (A : *) (x y z : A) (Hxy : eq A x y) (Hyz : eq A y z) : eq A x z := fun (P : A -> *) (Hx : P x) =>
    Hyz P (Hxy P Hx);
eq_cong (A B : *) (x y : A) (f : A -> B) (H : eq A x y) : eq B (f x) (f y) :=
  fun (P : B -> *) (Hfx : P (f x)) =>
      H (fun (a : A) => P (f a)) Hfx;

-- --------------------------------------------------------------------------------------------------------------
-- |                                        BASIC DEFINITIONS                                                   |
-- --------------------------------------------------------------------------------------------------------------

-- note we leave off the type ascriptions for most of these, as the type isn't
-- very interesting
-- I'd always strongly recommend including the type ascriptions for theorems

-- a unary operation on a set `A` is a function `A -> A`
unop (A : *) := A -> A;

-- a binary operation on a set `A` is a function `A -> A -> A`
binop (A : *) := A -> A -> A;

-- a binary operation is associative if ...
assoc (A : *) (op : binop A) :=
    forall (a b c : A), eq A (op (op a b) c) (op a (op b c));

-- an element `e : A` is a left identity with respect to binop `op` if `∀ a, e * a = a`
id_l (A : *) (op : binop A) (e : A) :=
    forall (a : A), eq A (op e a) a;

-- likewise for right identity
id_r (A : *) (op : binop A) (e : A) :=
    forall (a : A), eq A (op a e) a;

-- an element is an identity element if it is both a left and right identity
id (A : *) (op : binop A) (e : A) := and (id_l A op e) (id_r A op e);

-- b is a left inverse for a if `b * a = e`
-- NOTE: we don't require `e` to be an identity in this definition.
--       this definition is purely for convenience's sake
inv_l (A : *) (op : binop A) (e : A) (a b : A) := eq A (op b a) e;

-- likewise for right inverse
inv_r (A : *) (op : binop A) (e : A) (a b : A) := eq A (op a b) e;

-- and full-on inverse
inv (A : *) (op : binop A) (e : A) (a b : A) := and (inv_l A op e a b) (inv_r A op e a b);

-- --------------------------------------------------------------------------------------------------------------
-- |                                      ALGEBRAIC STRUCTURES                                                  |
-- --------------------------------------------------------------------------------------------------------------

-- a set `S` with binary operation `op` is a semigroup if its operation is associative
semigroup (S : *) (op : binop S) : * := assoc S op;

-- a set `M` with binary operation `op` and element `e` is a monoid 
monoid (M : *) (op : binop M) (e : M) : * :=
    and (semigroup M op) (id M op e);

-- some "getters" for `monoid` so we don't have to do a bunch of very verbose
-- and-eliminations every time we want to use something
id_lm (M : *) (op : binop M) (e : M) (Hmonoid : monoid M op e) : id_l M op e :=
    and_elim_l (id_l M op e) (id_r M op e)
        (and_elim_r (semigroup M op) (id M op e) Hmonoid);

id_rm (M : *) (op : binop M) (e : M) (Hmonoid : monoid M op e) : id_r M op e :=
    and_elim_r (id_l M op e) (id_r M op e)
        (and_elim_r (semigroup M op) (id M op e) Hmonoid);

-- now we can prove that, for any monoid, if `a` is a left identity, then it
-- must be "the" identity
monoid_id_l_implies_identity
    (M : *) (op : binop M) (e : M) (Hmonoid : monoid M op e)
    (a : M) (H : id_l M op a) : eq M a e :=
        -- WTS a = a * e = e
        -- we can use `eq_trans` to glue proofs of `a = a * e` and `a * e = e` together
        eq_trans M a (op a e) e

            -- first, `a = a * e`, but we'll use `eq_sym` to flip it around
            (eq_sym M (op a e) a
                -- now the goal is to show `a * e = a`, which follows immediately from `id_r`
                (id_rm M op e Hmonoid a))

            -- now we need to show that `a * e = e`, but this immediately follows from `H`
            (H e);

-- the analogous result for right identities
monoid_id_r_implies_identity
    (M : *) (op : binop M) (e : M) (Hmonoid : monoid M op e)
    (a : M) (H : id_r M op a) : eq M a e :=
        -- this time, we'll show `a = e * a = e`
        eq_trans M a (op e a) e

            -- first, `a = e * a`
            (eq_sym M (op e a) a
                -- this time, it immediately follows from `id_l`
                (id_lm M op e Hmonoid a))

            -- and `e * a = e`
            (H e);
  

-- groups are just monoids with inverses
group (G : *) (op : binop G) (e : G) (i : unop G) : * :=
    and (monoid G op e)
        (forall (a : G), inv G op e a (i a));