theorem revsnoc (a l: nat): $ rev (l |> a) = a : rev l $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqtr |
rev (l |> a) = rev (a : 0) ++ rev l -> rev (a : 0) ++ rev l = a : rev l -> rev (l |> a) = a : rev l |
| 2 |
|
revappend |
rev (l ++ a : 0) = rev (a : 0) ++ rev l |
| 3 |
2 |
conv snoc |
rev (l |> a) = rev (a : 0) ++ rev l |
| 4 |
1, 3 |
ax_mp |
rev (a : 0) ++ rev l = a : rev l -> rev (l |> a) = a : rev l |
| 5 |
|
eqtr |
rev (a : 0) ++ rev l = a : 0 ++ rev l -> a : 0 ++ rev l = a : rev l -> rev (a : 0) ++ rev l = a : rev l |
| 6 |
|
appendeq1 |
rev (a : 0) = a : 0 -> rev (a : 0) ++ rev l = a : 0 ++ rev l |
| 7 |
|
revsn |
rev (a : 0) = a : 0 |
| 8 |
6, 7 |
ax_mp |
rev (a : 0) ++ rev l = a : 0 ++ rev l |
| 9 |
5, 8 |
ax_mp |
a : 0 ++ rev l = a : rev l -> rev (a : 0) ++ rev l = a : rev l |
| 10 |
|
eqtr |
a : 0 ++ rev l = a : (0 ++ rev l) -> a : (0 ++ rev l) = a : rev l -> a : 0 ++ rev l = a : rev l |
| 11 |
|
appendS |
a : 0 ++ rev l = a : (0 ++ rev l) |
| 12 |
10, 11 |
ax_mp |
a : (0 ++ rev l) = a : rev l -> a : 0 ++ rev l = a : rev l |
| 13 |
|
conseq2 |
0 ++ rev l = rev l -> a : (0 ++ rev l) = a : rev l |
| 14 |
|
append0 |
0 ++ rev l = rev l |
| 15 |
13, 14 |
ax_mp |
a : (0 ++ rev l) = a : rev l |
| 16 |
12, 15 |
ax_mp |
a : 0 ++ rev l = a : rev l |
| 17 |
9, 16 |
ax_mp |
rev (a : 0) ++ rev l = a : rev l |
| 18 |
4, 17 |
ax_mp |
rev (l |> a) = a : rev l |
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)