theorem snoclen (a b: nat): $ len (a |> b) = suc (len a) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
len (a |> b) = len a + len (b : 0) -> len a + len (b : 0) = suc (len a) -> len (a |> b) = suc (len a) |
| 2 |
|
appendlen |
len (a ++ b : 0) = len a + len (b : 0) |
| 3 |
2 |
conv snoc |
len (a |> b) = len a + len (b : 0) |
| 4 |
1, 3 |
ax_mp |
len a + len (b : 0) = suc (len a) -> len (a |> b) = suc (len a) |
| 5 |
|
eqtr |
len a + len (b : 0) = len a + 1 -> len a + 1 = suc (len a) -> len a + len (b : 0) = suc (len a) |
| 6 |
|
addeq2 |
len (b : 0) = 1 -> len a + len (b : 0) = len a + 1 |
| 7 |
|
len1 |
len (b : 0) = 1 |
| 8 |
6, 7 |
ax_mp |
len a + len (b : 0) = len a + 1 |
| 9 |
5, 8 |
ax_mp |
len a + 1 = suc (len a) -> len a + len (b : 0) = suc (len a) |
| 10 |
|
add12 |
len a + 1 = suc (len a) |
| 11 |
9, 10 |
ax_mp |
len a + len (b : 0) = suc (len a) |
| 12 |
4, 11 |
ax_mp |
len (a |> b) = suc (len 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)