theorem shrshladdid (a b c: nat): $ c < 2 ^ b -> shr (shl a b + c) b = a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
muladddiv1lt |
2 ^ b != 0 /\ c < 2 ^ b -> (a * 2 ^ b + c) // 2 ^ b = a |
| 2 |
1 |
exp |
2 ^ b != 0 -> c < 2 ^ b -> (a * 2 ^ b + c) // 2 ^ b = a |
| 3 |
2 |
conv shl, shr |
2 ^ b != 0 -> c < 2 ^ b -> shr (shl a b + c) b = a |
| 4 |
|
pow2ne0 |
2 ^ b != 0 |
| 5 |
3, 4 |
ax_mp |
c < 2 ^ b -> shr (shl a b + c) b = a |
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)