theorem appendcan1 (a b c: nat): $ a ++ b = a ++ c <-> b = c $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr4 |
(a ++ b = a ++ c <-> a ++ b <= a ++ c /\ a ++ c <= a ++ b) -> (b = c <-> a ++ b <= a ++ c /\ a ++ c <= a ++ b) -> (a ++ b = a ++ c <-> b = c) |
| 2 |
|
eqlele |
a ++ b = a ++ c <-> a ++ b <= a ++ c /\ a ++ c <= a ++ b |
| 3 |
1, 2 |
ax_mp |
(b = c <-> a ++ b <= a ++ c /\ a ++ c <= a ++ b) -> (a ++ b = a ++ c <-> b = c) |
| 4 |
|
bitr |
(b = c <-> b <= c /\ c <= b) -> (b <= c /\ c <= b <-> a ++ b <= a ++ c /\ a ++ c <= a ++ b) -> (b = c <-> a ++ b <= a ++ c /\ a ++ c <= a ++ b) |
| 5 |
|
eqlele |
b = c <-> b <= c /\ c <= b |
| 6 |
4, 5 |
ax_mp |
(b <= c /\ c <= b <-> a ++ b <= a ++ c /\ a ++ c <= a ++ b) -> (b = c <-> a ++ b <= a ++ c /\ a ++ c <= a ++ b) |
| 7 |
|
aneq |
(b <= c <-> a ++ b <= a ++ c) -> (c <= b <-> a ++ c <= a ++ b) -> (b <= c /\ c <= b <-> a ++ b <= a ++ c /\ a ++ c <= a ++ b) |
| 8 |
|
leappend2 |
b <= c <-> a ++ b <= a ++ c |
| 9 |
7, 8 |
ax_mp |
(c <= b <-> a ++ c <= a ++ b) -> (b <= c /\ c <= b <-> a ++ b <= a ++ c /\ a ++ c <= a ++ b) |
| 10 |
|
leappend2 |
c <= b <-> a ++ c <= a ++ b |
| 11 |
9, 10 |
ax_mp |
b <= c /\ c <= b <-> a ++ b <= a ++ c /\ a ++ c <= a ++ b |
| 12 |
6, 11 |
ax_mp |
b = c <-> a ++ b <= a ++ c /\ a ++ c <= a ++ b |
| 13 |
3, 12 |
ax_mp |
a ++ b = a ++ c <-> b = 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)