pub theorem lrec0 (z: nat) (S: set): $ lrec z S 0 = z $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
lrec z S 0 = if (0 = 0) z (S @ (fst (0 - 1), snd (0 - 1), lrec z S (snd (0 - 1)))) ->
if (0 = 0) z (S @ (fst (0 - 1), snd (0 - 1), lrec z S (snd (0 - 1)))) = z ->
lrec z S 0 = z |
| 2 |
|
lrecval |
lrec z S 0 = if (0 = 0) z (S @ (fst (0 - 1), snd (0 - 1), lrec z S (snd (0 - 1)))) |
| 3 |
1, 2 |
ax_mp |
if (0 = 0) z (S @ (fst (0 - 1), snd (0 - 1), lrec z S (snd (0 - 1)))) = z -> lrec z S 0 = z |
| 4 |
|
ifpos |
0 = 0 -> if (0 = 0) z (S @ (fst (0 - 1), snd (0 - 1), lrec z S (snd (0 - 1)))) = z |
| 5 |
|
eqid |
0 = 0 |
| 6 |
4, 5 |
ax_mp |
if (0 = 0) z (S @ (fst (0 - 1), snd (0 - 1), lrec z S (snd (0 - 1)))) = z |
| 7 |
3, 6 |
ax_mp |
lrec 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)