theorem shrmodadd1 (a b c: nat): $ shr (a % 2 ^ (b + c)) b = shr a b % 2 ^ c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr4 |
shr (a % 2 ^ (b + c)) b = shr (a % (2 ^ b * 2 ^ c)) b -> shr a b % 2 ^ c = shr (a % (2 ^ b * 2 ^ c)) b -> shr (a % 2 ^ (b + c)) b = shr a b % 2 ^ c |
| 2 |
|
shreq1 |
a % 2 ^ (b + c) = a % (2 ^ b * 2 ^ c) -> shr (a % 2 ^ (b + c)) b = shr (a % (2 ^ b * 2 ^ c)) b |
| 3 |
|
modeq2 |
2 ^ (b + c) = 2 ^ b * 2 ^ c -> a % 2 ^ (b + c) = a % (2 ^ b * 2 ^ c) |
| 4 |
|
powadd |
2 ^ (b + c) = 2 ^ b * 2 ^ c |
| 5 |
3, 4 |
ax_mp |
a % 2 ^ (b + c) = a % (2 ^ b * 2 ^ c) |
| 6 |
2, 5 |
ax_mp |
shr (a % 2 ^ (b + c)) b = shr (a % (2 ^ b * 2 ^ c)) b |
| 7 |
1, 6 |
ax_mp |
shr a b % 2 ^ c = shr (a % (2 ^ b * 2 ^ c)) b -> shr (a % 2 ^ (b + c)) b = shr a b % 2 ^ c |
| 8 |
|
divmod1 |
a // 2 ^ b % 2 ^ c = a % (2 ^ b * 2 ^ c) // 2 ^ b |
| 9 |
8 |
conv shr |
shr a b % 2 ^ c = shr (a % (2 ^ b * 2 ^ c)) b |
| 10 |
7, 9 |
ax_mp |
shr (a % 2 ^ (b + c)) b = shr a b % 2 ^ c |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp,
itru),
axs_pred_calc
(ax_gen,
ax_4,
ax_5,
ax_6,
ax_7,
ax_10,
ax_11,
ax_12),
axs_set
(elab,
ax_8),
axs_the
(theid,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)