Theorem rlrec0 | index | src |

theorem rlrec0 (S: set) (z: nat): $ rlrec z S 0 = z $;
StepHypRefExpression
1 eqtr
rlrec z S 0 = lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) 0 -> lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) 0 = z -> rlrec z S 0 = z
2 lreceq3
rev 0 = 0 -> lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (rev 0) = lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) 0
3 2 conv rlrec
rev 0 = 0 -> rlrec z S 0 = lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) 0
4 rev0
rev 0 = 0
5 3, 4 ax_mp
rlrec z S 0 = lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) 0
6 1, 5 ax_mp
lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) 0 = z -> rlrec z S 0 = z
7 lrec0
lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) 0 = z
8 6, 7 ax_mp
rlrec z S 0 = z

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)