// 245II in YALE encoding

// Rules

D >< V => ;
B(a,b,c) >< Z => a~c, b~Z;
B(a,b,c) >< Lam(x,t,v) => b~Lam(x,t,w), v~DB(a,w,c);
DB(a,b,c) >< V => a~c, b~V;
C(x,y) >< Z => x~Z, y~Z;
C(x,y) >< Lam(s,t,u) =>
	 x~Lam(a,b,c), y~Lam(d,e,f),
	 s~Delta(a,d), t~Delta(b,e), u~C(c,f);
C(x,y) >< V => x~V, y~V;
Delta(x,y) >< Delta(a,b) => x~a, y~b;
B(x,a,b) >< Delta(c,d) => x~CB(a,b,Delta(c,d));
Delta(a,b) >< CB(x,y,z) =>
	x~Delta(c,d), y~Delta(e,f), z~Delta(B(a,c,e),B(b,d,f));
Delta(x,y) >< S(z) =>
	x~S(a), y~S(b), z~Delta(a,b);
Delta(x,y) >< Z => x~Z, y~Z;
Delta(x,y) >< V => x~V, y~V;
Delta(x,y) >< C(t,s) =>
	x~C(a,b), y~C(c,d), t~Delta(b,d), s~Delta(a,c);
Delta(x,y) >< AppTo(t,s) =>
	x~AppTo(a,b), y~AppTo(c,d), t~Delta(a,c), s~Delta(b,d);
Delta(x,y) >< Lam(t,s,u)=>
	x~Lam(a,b,c), y~Lam(d,e,f),
	t~Delta(a,d), s~Delta(b,e), u~Delta(c,f);
DB(x,y,z) >< CB(t,s,u) =>
	x~CB(d,e,u), y~CB(t,f,d), s~DB(e,f,z);
C(x,y) >< CB(t,s,u) =>
	x~CB(a,b,c), y~CB(d,e,f), 
	t~Delta(a,d), s~C(b,e), u~C(c,f);
D >< CB(a,b,c) => a~c, b~D;

Eps >< Eps => ;
Eps >< S(x) => x~Eps;
Eps >< V => ;
Eps >< D =>;
Eps >< Delta(t,s) => t~Eps,s~Eps;
Eps >< Lam(t,s,u) => t~Eps,s~Eps,u~Eps;
Eps >< AppTo(t,s) => t~Eps,s~Eps;
Eps >< B(t,s,u) => t~Eps,s~Eps,u~Eps;
Eps >< CB(t,s,u) => t~Eps,s~Eps,u~Eps;



Lam(x,t,c) >< AppTo(result,s) => result~t, x~s, c~D;
// We can write this avtive pair in nets as
//   result << AppTo(s, lam_term)  for  AppTo(result, s)~ lam_term.

// If the `<<' is defined as user macro, then it becomes more readable:
//   result << App(lam_term, s)  =def=  lam_term ~ App(result, s)


// Nets
id1 ~ Lam(x,x,V);
id2 ~ Lam(x,x,V);


// lam f. lam x. f(fx)
two ~ Lam(fb, Lam(x,r2,c),V),
  fb ~ B(c,f,V),
  f2,f1 << C(f),   // C(f2,f1) ~ f
  r2 << AppTo(r1, f2),
  r1 << AppTo(x, f1);   


// lam f. lam x. f(f(f(fx)))
four ~ Lam(fb, Lam(x, r4, c), V),
  fb~B(c, f, V),
  f4, C(f3, C(f2, f1)) << C(f),
  r4 << AppTo(r3, f4),
  r3 << AppTo(r2, f3),
  r2 << AppTo(r1, f2),
  r1 << AppTo(x, f1);


// lam f. lam x. f(f(f(f(fx))))
five ~ Lam(fb, Lam(x, r5, c), V),
  fb~B(c, f, V),
  f5, C(f4, C(f3, C(f2, f1))) << C(f),
  r5 << AppTo(r4, f5),
  r4 << AppTo(r3, f4),
  r3 << AppTo(r2, f3),
  r2 << AppTo(r1, f2),
  r1 << AppTo(x, f1);



// (((24)5)I)I
aux24 << AppTo(four, two);
aux245 << AppTo(five, aux24);
aux245I << AppTo(id1, aux245);
aux245II << AppTo(id2, aux245I);

aux245II;
exit;