Theorem srecval2 ≪ | index | src | ≫

theorem srecval2 (S: set) (n: nat): $ srec S n = S @ srecaux S n $;


srecaux S (suc n) @ n = srec S n -> srecaux S (suc n) @ n = S @ srecaux S n -> srec S n = S @ srecaux S n
n < suc n -> srecaux S (suc n) @ n = srec S n
n < suc n
srecaux S (suc n) @ n = srec S n
srecaux S (suc n) @ n = S @ srecaux S n -> srec S n = S @ srecaux S n
srecaux S (suc n) @ n = write (srecaux S n) n (S @ srecaux S n) @ n ->
  write (srecaux S n) n (S @ srecaux S n) @ n = S @ srecaux S n ->
  srecaux S (suc n) @ n = S @ srecaux S n
srecaux S (suc n) == write (srecaux S n) n (S @ srecaux S n) -> srecaux S (suc n) @ n = write (srecaux S n) n (S @ srecaux S n) @ n
srecaux S (suc n) == write (srecaux S n) n (S @ srecaux S n)
srecaux S (suc n) @ n = write (srecaux S n) n (S @ srecaux S n) @ n
write (srecaux S n) n (S @ srecaux S n) @ n = S @ srecaux S n -> srecaux S (suc n) @ n = S @ srecaux S n
write (srecaux S n) n (S @ srecaux S n) @ n = S @ srecaux S n
srecaux S (suc n) @ n = S @ srecaux S n
srec S n = S @ srecaux S n

Step	Hyp	Ref	Expression
1		eqtr3	srecaux S (suc n) @ n = srec S n -> srecaux S (suc n) @ n = S @ srecaux S n -> srec S n = S @ srecaux S n
2		srecauxapp	n < suc n -> srecaux S (suc n) @ n = srec S n
3		ltsucid	n < suc n
4	2, 3	ax_mp	srecaux S (suc n) @ n = srec S n
5	1, 4	ax_mp	srecaux S (suc n) @ n = S @ srecaux S n -> srec S n = S @ srecaux S n
6		eqtr	srecaux S (suc n) @ n = write (srecaux S n) n (S @ srecaux S n) @ n -> write (srecaux S n) n (S @ srecaux S n) @ n = S @ srecaux S n -> srecaux S (suc n) @ n = S @ srecaux S n
7		appeq1	srecaux S (suc n) == write (srecaux S n) n (S @ srecaux S n) -> srecaux S (suc n) @ n = write (srecaux S n) n (S @ srecaux S n) @ n
8		srecauxS	srecaux S (suc n) == write (srecaux S n) n (S @ srecaux S n)
9	7, 8	ax_mp	srecaux S (suc n) @ n = write (srecaux S n) n (S @ srecaux S n) @ n
10	6, 9	ax_mp	write (srecaux S n) n (S @ srecaux S n) @ n = S @ srecaux S n -> srecaux S (suc n) @ n = S @ srecaux S n
11		writeEq	write (srecaux S n) n (S @ srecaux S n) @ n = S @ srecaux S n
12	10, 11	ax_mp	srecaux S (suc n) @ n = S @ srecaux S n
13	5, 12	ax_mp	srec S n = S @ srecaux S n

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)