theorem b1ltb0 (a b: nat): $ b1 a < b0 b <-> a < b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(b1 a < b0 b <-> suc (b1 a) < suc (b0 b)) -> (suc (b1 a) < suc (b0 b) <-> a < b) -> (b1 a < b0 b <-> a < b) |
| 2 |
|
ltsuc |
b1 a < b0 b <-> suc (b1 a) < suc (b0 b) |
| 3 |
1, 2 |
ax_mp |
(suc (b1 a) < suc (b0 b) <-> a < b) -> (b1 a < b0 b <-> a < b) |
| 4 |
|
bitr |
(suc (b1 a) < suc (b0 b) <-> b0 (suc a) < b1 b) -> (b0 (suc a) < b1 b <-> a < b) -> (suc (b1 a) < suc (b0 b) <-> a < b) |
| 5 |
|
lteq |
suc (b1 a) = b0 (suc a) -> suc (b0 b) = b1 b -> (suc (b1 a) < suc (b0 b) <-> b0 (suc a) < b1 b) |
| 6 |
|
sucb1 |
suc (b1 a) = b0 (suc a) |
| 7 |
5, 6 |
ax_mp |
suc (b0 b) = b1 b -> (suc (b1 a) < suc (b0 b) <-> b0 (suc a) < b1 b) |
| 8 |
|
sucb0 |
suc (b0 b) = b1 b |
| 9 |
7, 8 |
ax_mp |
suc (b1 a) < suc (b0 b) <-> b0 (suc a) < b1 b |
| 10 |
4, 9 |
ax_mp |
(b0 (suc a) < b1 b <-> a < b) -> (suc (b1 a) < suc (b0 b) <-> a < b) |
| 11 |
|
b0ltb1 |
b0 (suc a) < b1 b <-> suc a <= b |
| 12 |
11 |
conv lt |
b0 (suc a) < b1 b <-> a < b |
| 13 |
10, 12 |
ax_mp |
suc (b1 a) < suc (b0 b) <-> a < b |
| 14 |
3, 13 |
ax_mp |
b1 a < b0 b <-> a < 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
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)