theorem shrshl1 (a b c: nat): $ c <= b -> shr (shl a b) c = shl a (b - c) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
shrshlid |
shr (shl (shl a (b - c)) c) c = shl a (b - c) |
| 2 |
|
shlshl |
shl (shl a (b - c)) c = shl a (b - c + c) |
| 3 |
|
npcan |
c <= b -> b - c + c = b |
| 4 |
3 |
eqcomd |
c <= b -> b = b - c + c |
| 5 |
4 |
shleq2d |
c <= b -> shl a b = shl a (b - c + c) |
| 6 |
2, 5 |
syl6eqr |
c <= b -> shl a b = shl (shl a (b - c)) c |
| 7 |
6 |
shreq1d |
c <= b -> shr (shl a b) c = shr (shl (shl a (b - c)) c) c |
| 8 |
1, 7 |
syl6eq |
c <= b -> shr (shl a b) c = shl a (b - 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)