-- 
-- We need some basic logic, so I'm stealing this from <logic.pg>. See that file
-- for more details on how these work.
-- 

@include logic.pg

binop (A : *) := A -> A -> A;

comp (A B C : *) (g : B -> C) (f : A -> B) (x : A) : C :=
  g (f x);

assoc (A : *) (op : binop A) := forall (c a b : A),
  eq A (op a (op b c)) (op (op a b) c);

-- --------------------------------------------------------------------------------------------------------------

-- 
-- Now we can define the Peano axioms. Unlike with equality, perga is not
-- powerful enough to construct the natural numbers (or at least to prove the
-- Peano axioms as theorems from a definition constructible in perga). However,
-- working with axioms is extremely common in math. As such, perga has a system
-- for doing just that, namely the *axiom* system.
-- 
-- In a definition, rather than give a value for the term, we can give it the
-- value axiom, in which case a type ascription is mandatory. Perga will then
-- trust our type ascription, and assume going forward that the identifier we
-- defined is a value of the asserted type. For example, we will use the axiom
-- system to assert the existence of a type of natural numbers
-- 

nat : * := axiom;

-- 
-- As you can imagine, this can be risky. For instance, there's nothing stopping
-- us from saying
--     uh_oh : false := axiom;
-- or stipulating more subtly contradictory axioms. As such, as in mathematics,
-- axioms should be used with care.
-- 
-- There's another problem with axioms, namely that perga cannot do any
-- computations with axioms. The more you can define within perga natively, the
-- better, as computations done without axioms can be utilized by perga. For
-- example, in <computation.pg>, we define the natural numbers as Church
-- numerals entirely within perga. There, the proof that 1 + 1 = 2 is just
-- eq_refl, since they reduce to the same thing. Here, 1 + 1 = 2 will
-- require a proof, since perga is unable to do computations with things defined
-- as axioms.
-- 
-- With these warnings in place, the Peano axioms are proven to be consistent,
-- so we should be fine. I'm formalizing the second order axioms given in the
-- wikipedia article on the Peano axioms
-- (https://en.wikipedia.org/wiki/Peano_axioms).
-- 

-- axiom 1: 0 is a natural number
zero : nat := axiom;

-- axiom 6: For every n, S n is a natural number.
suc (n : nat) : nat := axiom;

-- axiom 7: If S n = S m, then n = m.
suc_inj : forall (n m : nat), eq nat (suc n) (suc m) -> eq nat n m := axiom;

-- axiom 8: No successor of any natural number is zero.
suc_nonzero : forall (n : nat), not (eq nat (suc n) zero) := axiom;

-- axiom 9: Induction! For any proposition φ on natural numbers, if φ(0) holds,
-- and if for every natural number n, φ(n) ⇒ φ(S n), then φ holds for all n.
nat_ind : forall (φ : nat -> *), φ zero -> (forall (n : nat), φ n -> φ (suc n))
    -> forall (n : nat), φ n := axiom;

-- --------------------------------------------------------------------------------------------------------------

-- 
-- Now that we have stipulated these axioms, we are free to use them to make
-- definitions, prove theorems, etc.
-- 
-- Our first theorem, as a warm up, is to prove that every natural number is
-- either 0 or the successor of another natural number.
-- 
-- First, we will make a bunch of abbreviations, since these terms get really
-- long and complicated really quickly.
-- 

-- Some abbreviations for common numbers.
one   : nat := suc zero;
two   : nat := suc one;
three : nat := suc two;
four  : nat := suc three;
five  : nat := suc four;

-- First, the predecessor of n is m if n = suc m.
pred (n m : nat) : * := eq nat n (suc m);

-- Our claim is a disjunction, whose first option is that n = 0.
szc_l (n : nat) := eq nat n zero;

-- The second option is that n has a predecessor.
szc_r (n : nat) := exists nat (pred n);

-- So the claim we are trying to prove is that either one of the above options
-- holds for every n.
szc (n : nat) := or (szc_l n) (szc_r n);

-- And here's our proof!
suc_or_zero : forall (n : nat), szc n :=
    -- We will prove this by induction.
    nat_ind szc
        -- For the base case, the first option holds, i.e. 0 = 0
        (or_intro_l (szc_l zero) (szc_r zero) (eq_refl nat zero))

        -- For the inductive case, suppose the theorem holds for n.
        (fun (n : nat) (_ : szc n) => 
            -- Then the right option holds for suc n, since suc n is the
            -- successor of n
            or_intro_r (szc_l (suc n)) (szc_r (suc n))
                (exists_intro nat (pred (suc n)) n (eq_refl nat (suc n))));

-- 
-- Before we continue one to defining addition, we need one more bunch of
-- axioms. Addition is normally defined recursively, which is difficult without
-- inductive definitions. I'm pretty sure this axiom is derivable from the
-- others, but I've spent a very long time trying to derive it, and it has
-- proven extremely difficult. I'm still convinced it's possible, just
-- prohibitively difficult and tedious without further language features
-- (implicit arguments and let bindings would help a ton).
-- 
-- Normally, we define recursive functions f : nat -> A in the following format:
--            f 0 = fzero
--      f (suc n) = fsuc n (f n)
-- More concretely, we have two equations, one for each possible case for n,
-- i.e. either zero or a successor. In the successor case, the value of
-- f (suc n) can depend on the value of f n. The recursion axioms below
-- assert that for any such pair of equations, there exists a unique function
-- satisfying said equations.
-- 

-- For any such equations, there exists a function.
rec_def (A : *) (fzero : A) (fsuc : nat -> A -> A) : nat -> A := axiom;

-- Here's equation one.
rec_cond_zero (A : *) (fzero : A) (fsuc : nat -> A -> A) (f : nat -> A) := 
  eq A (f zero) fzero;

-- And equation two.
rec_cond_suc (A : *) (fzero : A) (fsuc : nat -> A -> A) (f : nat -> A) :=
  forall (n : nat) (y : A),
    eq A (f n) y -> eq A (f (suc n)) (fsuc n y);

-- Said function satisfies the equations.
-- It satisfies equation one.
rec_def_sat_zero (A : *) (fzero : A) (fsuc : nat -> A -> A) :
  rec_cond_zero A fzero fsuc (rec_def A fzero fsuc) := axiom;

-- And two.
rec_def_sat_suc (A : *) (fzero : A) (fsuc : nat -> A -> A) :
  rec_cond_suc A fzero fsuc (rec_def A fzero fsuc) := axiom;

-- And, finally, this function is unique in the sense that if any other function
-- also satisfies the equations, it takes the same values as rec_def.
rec_def_unique (A : *) (fzero : A) (fsuc : nat -> A -> A) (f g : nat -> A) :
  rec_cond_zero A fzero fsuc f ->
  rec_cond_suc  A fzero fsuc f ->
  rec_cond_zero A fzero fsuc g ->
  rec_cond_suc  A fzero fsuc g ->
  forall (x : nat), eq A (f x) (g x) := axiom;

-- Now we can safely define addition.

-- First, here's the RHS of equation 2 as a function, since it will show up
-- multiple times.
psuc (_ r : nat) := suc r;

-- And here's plus!
plus (n : nat) : nat -> nat := rec_def nat n psuc;

-- The first equation manifests itself as the familiar
--      n + 0 = 0.
plus_0_r (n : nat) : eq nat (plus n zero) n :=
  rec_def_sat_zero nat n psuc;

-- The second equation, after a bit of massaging, manifests itself as the
-- likewise familiar
--      n + suc m = suc (n + m).
plus_s_r (n m : nat) : eq nat (plus n (suc m)) (suc (plus n m)) :=
  rec_def_sat_suc nat n psuc m (plus n m) (eq_refl nat (plus n m));

-- We can now prove 1 + 1 = 2!
one_plus_one_two : eq nat (plus one one) two :=
  -- 1 + (suc zero) = suc (1 + zero) = suc one
  eq_trans nat (plus one one) (suc (plus one zero)) two

  -- 1 + (suc zero) = suc (1 + zero)
  (plus_s_r one zero)

  -- suc (1 + zero) = suc one
  (eq_cong nat nat (plus one zero) one suc (plus_0_r one));

-- 
-- We have successfully defined addition! Note that evaluating 1 + 1 to 2
-- requires a proof, unfortunately, since this computation isn't visible to
-- perga.
-- 
-- We will now prove a couple standard properties of addition.
-- 

-- First, associativity, namely that n + (m + p) = (n + m) + p.
plus_assoc : assoc nat plus := 
  -- We prove this via induction on p.
  nat_ind
    (fun (p : nat) =>
      forall (n m : nat),
        eq nat (plus n (plus m p)) (plus (plus n m) p))

    -- Base case: p = 0
    -- WTS n + (m + 0) = (n + m) + 0
    (fun (n m : nat) =>
      -- n + (m + 0) = n + m = (n + m) + 0
      (eq_trans nat (plus n (plus m zero)) (plus n m) (plus (plus n m) zero)
        -- n + (m + 0) = n + m
        (eq_cong nat nat (plus m zero) m (fun (p : nat) => plus n p) (plus_0_r m))

        -- n + m = (n + m) + 0
        (eq_sym nat (plus (plus n m) zero) (plus n m) (plus_0_r (plus n m)))))

    -- Inductive step: IH = n + (m + p) = (n + m) + p
    (fun (p : nat) (IH : forall (n m : nat), eq nat (plus n (plus m p)) (plus (plus n m) p)) (n m : nat) =>
      -- WTS n + (m + suc p) = (n + m) + suc p
      -- n + (m + suc p) = n + suc (m + p)
      --                 = suc (n + (m + p))
      --                 = suc ((n + m) + p)
      --                 = (n + m) + suc p
      eq_trans nat (plus n (plus m (suc p))) (plus n (suc (plus m p))) (plus (plus n m) (suc p))

      -- n + (m + suc p) = n + suc (m + p)
      (eq_cong nat nat (plus m (suc p)) (suc (plus m p)) (fun (a : nat) => (plus n a)) (plus_s_r m p))

      -- n + suc (m + p) = (n + m) + suc p
      (eq_trans nat (plus n (suc (plus m p))) (suc (plus n (plus m p))) (plus (plus n m) (suc p))
        -- n + suc (m + p) = suc (n + (m + p))
        (plus_s_r n (plus m p))

        -- suc (n + (m + p)) = (n + m) + suc p
        (eq_trans nat (suc (plus n (plus m p))) (suc (plus (plus n m) p)) (plus (plus n m) (suc p))
          -- suc (n + (m + p)) = suc ((n + m) + p)
          (eq_cong nat nat (plus n (plus m p)) (plus (plus n m) p) suc (IH n m))

          -- suc ((n + m) + p) = (n + m) + suc p
          (eq_sym nat (plus (plus n m) (suc p)) (suc (plus (plus n m) p))
            (plus_s_r (plus n m) p)))));

-- Up next is commutativity, but we will need a couple lemmas first.

-- First, we will show that 0 + n = n.
plus_0_l : forall (n : nat), eq nat (plus zero n) n :=
  -- We prove this by induction on n.
  nat_ind (fun (n : nat) => eq nat (plus zero n) n)
    -- base case: WTS 0 + 0 = 0
    -- This is just plus_0_r 0
    (plus_0_r zero)

    -- inductive case
    (fun (n : nat) (IH : eq nat (plus zero n) n) =>
      -- WTS 0 + (suc n) = suc n
      -- 0 + (suc n) = suc (0 + n) = suc n
      eq_trans nat (plus zero (suc n)) (suc (plus zero n)) (suc n)
      -- 0 + (suc n) = suc (0 + n)
      (plus_s_r zero n)

      -- suc (0 + n) = suc n
      (eq_cong nat nat (plus zero n) n suc IH));

-- Next, we will show that (suc n) + m = suc (n + m).
plus_s_l (n : nat) : forall (m : nat), eq nat (plus (suc n) m) (suc (plus n m)) :=
  -- We proceed by induction on m.
  nat_ind (fun (m : nat) => eq nat (plus (suc n) m) (suc (plus n m)))
    -- base case: (suc n) + 0 = suc (n + 0)
    -- (suc n) + 0 = suc n = suc (n + 0)
    (eq_trans nat (plus (suc n) zero) (suc n) (suc (plus n zero))
      -- (suc n) + 0 = suc n
      (plus_0_r (suc n))

      -- suc n = suc (n + 0)
      (eq_cong nat nat n (plus n zero) suc
        -- n = n + 0
        (eq_sym nat (plus n zero) n (plus_0_r n))))

    -- inductive case
    -- IH = suc n + m = suc (n + m)
    (fun (m : nat) (IH : eq nat (plus (suc n) m) (suc (plus n m))) =>
        -- WTS suc n + suc m = suc (n + suc m)
        -- suc n + suc m = suc (suc n + m)
        --               = suc (suc (n + m))
        --               = suc (n + suc m)
        (eq_trans nat (plus (suc n) (suc m)) (suc (plus (suc n) m)) (suc (plus n (suc m)))
          -- suc n + suc m = suc (suc n + m)
          (plus_s_r (suc n) m)

          -- suc (suc n + m) = suc (n + suc m)
          (eq_trans nat (suc (plus (suc n) m)) (suc (suc (plus n m))) (suc (plus n (suc m)))
            -- suc (suc n + m) = suc (suc (n + m))
            (eq_cong nat nat (plus (suc n) m) (suc (plus n m)) suc IH)

            -- suc (suc (n + m)) = suc (n + suc m)
            (eq_cong nat nat (suc (plus n m)) (plus n (suc m)) suc
              -- suc (n + m) = n + suc m
              (eq_sym nat (plus n (suc m)) (suc (plus n m)) (plus_s_r n m))))));

-- Finally, we can prove commutativity.
plus_comm (n : nat) : forall (m : nat), eq nat (plus n m) (plus m n) :=
  -- As usual, we proceed by induction.
  nat_ind (fun (m : nat) => eq nat (plus n m) (plus m n))

  -- Base case: WTS n + 0 = 0 + n
  -- n + 0 = n = 0 + n
  (eq_trans nat (plus n zero) n (plus zero n)
    -- n + 0 = n
    (plus_0_r n)

    -- n = 0 + n
    (eq_sym nat (plus zero n) n (plus_0_l n)))

  -- Inductive step:
  (fun (m : nat) (IH : eq nat (plus n m) (plus m n)) =>
    -- WTS n + suc m = suc m + n
    -- n + suc m = suc (n + m)
    --           = suc (m + n)
    --           = suc m + n
    (eq_trans nat (plus n (suc m)) (suc (plus n m)) (plus (suc m) n)
      -- n + suc m = suc (n + m)
      (plus_s_r n m)

      -- suc (n + m) = suc m + n
      (eq_trans nat (suc (plus n m)) (suc (plus m n)) (plus (suc m) n)
        -- suc (n + m) = suc (m + n)
        (eq_cong nat nat (plus n m) (plus m n) suc IH)

        -- suc (m + n) = suc m + n
        (eq_sym nat (plus (suc m) n) (suc (plus m n)) (plus_s_l m n)))));