theorem sublistAt_id (L: nat): $ sublistAt 0 L L $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
ian |
L = 0 ++ L ++ 0 -> len 0 = 0 -> L = 0 ++ L ++ 0 /\ len 0 = 0 |
| 2 |
|
eqtr2 |
0 ++ L ++ 0 = L ++ 0 -> L ++ 0 = L -> L = 0 ++ L ++ 0 |
| 3 |
|
append0 |
0 ++ L ++ 0 = L ++ 0 |
| 4 |
2, 3 |
ax_mp |
L ++ 0 = L -> L = 0 ++ L ++ 0 |
| 5 |
|
append02 |
L ++ 0 = L |
| 6 |
4, 5 |
ax_mp |
L = 0 ++ L ++ 0 |
| 7 |
1, 6 |
ax_mp |
len 0 = 0 -> L = 0 ++ L ++ 0 /\ len 0 = 0 |
| 8 |
|
len0 |
len 0 = 0 |
| 9 |
7, 8 |
ax_mp |
L = 0 ++ L ++ 0 /\ len 0 = 0 |
| 10 |
|
id |
x = 0 -> x = 0 |
| 11 |
10 |
anwl |
x = 0 /\ y = 0 -> x = 0 |
| 12 |
|
id |
y = 0 -> y = 0 |
| 13 |
12 |
anwr |
x = 0 /\ y = 0 -> y = 0 |
| 14 |
13 |
appendeq2d |
x = 0 /\ y = 0 -> L ++ y = L ++ 0 |
| 15 |
11, 14 |
appendeqd |
x = 0 /\ y = 0 -> x ++ L ++ y = 0 ++ L ++ 0 |
| 16 |
15 |
eqeq2d |
x = 0 /\ y = 0 -> (L = x ++ L ++ y <-> L = 0 ++ L ++ 0) |
| 17 |
11 |
leneqd |
x = 0 /\ y = 0 -> len x = len 0 |
| 18 |
17 |
eqeq1d |
x = 0 /\ y = 0 -> (len x = 0 <-> len 0 = 0) |
| 19 |
16, 18 |
aneqd |
x = 0 /\ y = 0 -> (L = x ++ L ++ y /\ len x = 0 <-> L = 0 ++ L ++ 0 /\ len 0 = 0) |
| 20 |
9, 19 |
mpbiri |
x = 0 /\ y = 0 -> L = x ++ L ++ y /\ len x = 0 |
| 21 |
20 |
iexde |
x = 0 -> E. y (L = x ++ L ++ y /\ len x = 0) |
| 22 |
21 |
iexie |
E. x E. y (L = x ++ L ++ y /\ len x = 0) |
| 23 |
22 |
conv sublistAt |
sublistAt 0 L 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)