theorem lesnoc2 (a b c: nat): $ b <= c <-> a |> b <= a |> c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(b <= c <-> b : 0 <= c : 0) -> (b : 0 <= c : 0 <-> a |> b <= a |> c) -> (b <= c <-> a |> b <= a |> c) |
| 2 |
|
lecons1 |
b <= c <-> b : 0 <= c : 0 |
| 3 |
1, 2 |
ax_mp |
(b : 0 <= c : 0 <-> a |> b <= a |> c) -> (b <= c <-> a |> b <= a |> c) |
| 4 |
|
leappend2 |
b : 0 <= c : 0 <-> a ++ b : 0 <= a ++ c : 0 |
| 5 |
4 |
conv snoc |
b : 0 <= c : 0 <-> a |> b <= a |> c |
| 6 |
3, 5 |
ax_mp |
b <= c <-> a |> b <= a |> c |
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)