theorem lesuc (a b: nat): $ a <= b <-> suc a <= suc b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(a <= b <-> a + 1 <= b + 1) -> (a + 1 <= b + 1 <-> suc a <= suc b) -> (a <= b <-> suc a <= suc b) |
| 2 |
|
leadd1 |
a <= b <-> a + 1 <= b + 1 |
| 3 |
1, 2 |
ax_mp |
(a + 1 <= b + 1 <-> suc a <= suc b) -> (a <= b <-> suc a <= suc b) |
| 4 |
|
leeq |
a + 1 = suc a -> b + 1 = suc b -> (a + 1 <= b + 1 <-> suc a <= suc b) |
| 5 |
|
add12 |
a + 1 = suc a |
| 6 |
4, 5 |
ax_mp |
b + 1 = suc b -> (a + 1 <= b + 1 <-> suc a <= suc b) |
| 7 |
|
add12 |
b + 1 = suc b |
| 8 |
6, 7 |
ax_mp |
a + 1 <= b + 1 <-> suc a <= suc b |
| 9 |
3, 8 |
ax_mp |
a <= b <-> suc a <= suc b |
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_peano
(peano2,
peano5,
addeq,
add0,
addS)