pub theorem nthZ (a l: nat): $ nth 0 (a : l) = suc a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
nth 0 (a : l) = suc (listfn (a : l) @ 0) -> suc (listfn (a : l) @ 0) = suc a -> nth 0 (a : l) = suc a |
| 2 |
|
ifpos |
0 < len (a : l) -> if (0 < len (a : l)) (suc (listfn (a : l) @ 0)) 0 = suc (listfn (a : l) @ 0) |
| 3 |
2 |
conv nth |
0 < len (a : l) -> nth 0 (a : l) = suc (listfn (a : l) @ 0) |
| 4 |
|
lteq2 |
len (a : l) = suc (len l) -> (0 < len (a : l) <-> 0 < suc (len l)) |
| 5 |
|
lenS |
len (a : l) = suc (len l) |
| 6 |
4, 5 |
ax_mp |
0 < len (a : l) <-> 0 < suc (len l) |
| 7 |
|
lt01S |
0 < suc (len l) |
| 8 |
6, 7 |
mpbir |
0 < len (a : l) |
| 9 |
3, 8 |
ax_mp |
nth 0 (a : l) = suc (listfn (a : l) @ 0) |
| 10 |
1, 9 |
ax_mp |
suc (listfn (a : l) @ 0) = suc a -> nth 0 (a : l) = suc a |
| 11 |
|
suceq |
listfn (a : l) @ 0 = a -> suc (listfn (a : l) @ 0) = suc a |
| 12 |
|
listfnS0 |
listfn (a : l) @ 0 = a |
| 13 |
11, 12 |
ax_mp |
suc (listfn (a : l) @ 0) = suc a |
| 14 |
10, 13 |
ax_mp |
nth 0 (a : l) = suc 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)