1	# bend-osdev-word64's laws of division, for every input: W64.div and
2	# W64.mod are the Nat quotient and remainder, and division by 0 gives 0 and
3	# the dividend. Proved in PROOF_div.bend.
4	import Base
5	import bend-osdev-u32@0.1.0.0/u32.bend as U
6	import ./word64.bend as W
7	
8	# Division (shift and subtract, 64 steps): for b not 0, the quotient and the
9	# remainder are the Nat ones, a = q b + r with r < b; by 0, the quotient is 0
10	# and the remainder a.
11	law divmod_value:
12	  for +a: W.W64
13	  for +b: W.W64
14	  for h: {W.W64.is_zero(b) == False{} : Bool}
15	  {Nat.add(Nat.mul(W.W64.to_nat(W.W64.div(a, b)), W.W64.to_nat(b)), W.W64.to_nat(W.W64.mod(a, b))) == W.W64.to_nat(a) : Nat}
16	
17	law mod_lt:
18	  for +a: W.W64
19	  for +b: W.W64
20	  for h: {W.W64.is_zero(b) == False{} : Bool}
21	  {Nat.is_lt(W.W64.to_nat(W.W64.mod(a, b)), W.W64.to_nat(b)) == True{} : Bool}
22	
23	law div_zero:
24	  for +a: W.W64
25	  {W.W64.div(a, W.W64.zero()) == W.W64.zero() : W.W64}
26	
27	law mod_zero:
28	  for +a: W.W64
29	  {W.W64.mod(a, W.W64.zero()) == a : W.W64}