Theorem nth0 ≪ | index | src | ≫

pub theorem nth0 (n: nat): $ nth n 0 = 0 $;

Step	Hyp	Ref	Expression
1		ifneg	~n < len 0 -> if (n < len 0) (suc (listfn 0 @ n)) 0 = 0
2	1	conv nth	~n < len 0 -> nth n 0 = 0
3		lteq2	len 0 = 0 -> (n < len 0 <-> n < 0)
4		len0	len 0 = 0
5	3, 4	ax_mp	n < len 0 <-> n < 0
6		lt02	~n < 0
7	5, 6	mtbir	~n < len 0
8	2, 7	ax_mp	nth n 0 = 0

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)