theorem cardb1 (n: nat): $ card (b1 n) = suc (card n) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
card (b1 n) = card (b1 n // 2) + b1 n % 2 -> card (b1 n // 2) + b1 n % 2 = suc (card n) -> card (b1 n) = suc (card n) |
| 2 |
|
cardval |
b1 n != 0 -> card (b1 n) = card (b1 n // 2) + b1 n % 2 |
| 3 |
|
b1ne0 |
b1 n != 0 |
| 4 |
2, 3 |
ax_mp |
card (b1 n) = card (b1 n // 2) + b1 n % 2 |
| 5 |
1, 4 |
ax_mp |
card (b1 n // 2) + b1 n % 2 = suc (card n) -> card (b1 n) = suc (card n) |
| 6 |
|
eqtr |
card (b1 n // 2) + b1 n % 2 = card n + 1 -> card n + 1 = suc (card n) -> card (b1 n // 2) + b1 n % 2 = suc (card n) |
| 7 |
|
addeq |
card (b1 n // 2) = card n -> b1 n % 2 = 1 -> card (b1 n // 2) + b1 n % 2 = card n + 1 |
| 8 |
|
cardeq |
b1 n // 2 = n -> card (b1 n // 2) = card n |
| 9 |
|
b1div2 |
b1 n // 2 = n |
| 10 |
8, 9 |
ax_mp |
card (b1 n // 2) = card n |
| 11 |
7, 10 |
ax_mp |
b1 n % 2 = 1 -> card (b1 n // 2) + b1 n % 2 = card n + 1 |
| 12 |
|
b1mod2 |
b1 n % 2 = 1 |
| 13 |
11, 12 |
ax_mp |
card (b1 n // 2) + b1 n % 2 = card n + 1 |
| 14 |
6, 13 |
ax_mp |
card n + 1 = suc (card n) -> card (b1 n // 2) + b1 n % 2 = suc (card n) |
| 15 |
|
add12 |
card n + 1 = suc (card n) |
| 16 |
14, 15 |
ax_mp |
card (b1 n // 2) + b1 n % 2 = suc (card n) |
| 17 |
5, 16 |
ax_mp |
card (b1 n) = suc (card n) |
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)