theorem recn0 (S: set) (z: nat): $ recn z S 0 = z $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
recn z S 0 = snd (0, z) -> snd (0, z) = z -> recn z S 0 = z |
| 2 |
|
sndeq |
recnaux z S 0 = 0, z -> snd (recnaux z S 0) = snd (0, z) |
| 3 |
2 |
conv recn |
recnaux z S 0 = 0, z -> recn z S 0 = snd (0, z) |
| 4 |
|
rec0 |
rec (0, z) (\ a1, suc (fst a1), S @ a1) 0 = 0, z |
| 5 |
4 |
conv recnaux |
recnaux z S 0 = 0, z |
| 6 |
3, 5 |
ax_mp |
recn z S 0 = snd (0, z) |
| 7 |
1, 6 |
ax_mp |
snd (0, z) = z -> recn z S 0 = z |
| 8 |
|
sndpr |
snd (0, z) = z |
| 9 |
7, 8 |
ax_mp |
recn z S 0 = z |
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)