% Definition of plus
val plus : nat -> nat -> nat =
fn x => rec x of 
   p 0 => fn y => y
 | p (s n) => fn y => s (p n y)
 end;

% --------------------------------------------------------------------
% Tutch proofs

% Here's an example of a proof that uses induction over natural numbers.
proof plus0R : (!m:nat. m = plus m 0) =
begin
[m : nat;
 
 % the 0 case
 0 = plus 0 0;

 % the s case; note that it binds two assumptions
 [x : nat , x = plus x 0;
  s x = plus (s x) 0];

 m = plus m 0];
(!m:nat. m = plus m 0);
end;

% Your task is to prove transitivity
% Hint: this will be longer than any tutch proof you have done so far
proof eqtrans : !x:nat. !y:nat. !z:nat. (x = y) => (y = z) => (x = z) =
begin
!x:nat. !y:nat. !z:nat. (x = y) => (y = z) => (x = z);
end;

% ----------------------------------------------------------------------
% Proof terms

% Here's an example proof term using recursion.  The recursor
%    R(t,M0,x.u.M1(x,u)) 
% is written as a primitive recursion schema:
%    rec t of f 0 => M0 
%           | f (s x) => M1(x,f(x)) end;

term plus0R : (!m:nat. m = plus m 0) =
fn m => rec m of 
          p 0 => eq0
        | p (s x) => eqS (p x)
end; 


% Prove the following:

term eqrefl : (!m:nat. m = m) = 
fn m => m; 

% Hint: you will need to call the function eqrefl in this proof
term plusSR : (!m:nat. !n:nat. s (plus m n) = plus m (s n)) =
fn m => m;