Theorem recnauxS2 ≪ | index | src | ≫ 

theorem recnauxS2 (S: set) (n z: nat):
  $ recnaux z S (suc n) = suc (fst (recnaux z S n)), S @ recnaux z S n $;
StepHypRefExpression
1 eqtr 
recnaux z S (suc n) = (\ p, suc (fst p), S @ p) @ rec (0, z) (\ p, suc (fst p), S @ p) n ->
  (\ p, suc (fst p), S @ p) @ rec (0, z) (\ p, suc (fst p), S @ p) n = suc (fst (recnaux z S n)), S @ recnaux z S n ->
  recnaux z S (suc n) = suc (fst (recnaux z S n)), S @ recnaux z S n
2 recS 
rec (0, z) (\ p, suc (fst p), S @ p) (suc n) = (\ p, suc (fst p), S @ p) @ rec (0, z) (\ p, suc (fst p), S @ p) n
3 2 conv recnaux 
recnaux z S (suc n) = (\ p, suc (fst p), S @ p) @ rec (0, z) (\ p, suc (fst p), S @ p) n
4 1, 3 ax_mp 
(\ p, suc (fst p), S @ p) @ rec (0, z) (\ p, suc (fst p), S @ p) n = suc (fst (recnaux z S n)), S @ recnaux z S n ->
  recnaux z S (suc n) = suc (fst (recnaux z S n)), S @ recnaux z S n
5 fsteq 
p = recnaux z S n -> fst p = fst (recnaux z S n)
6 5 suceqd 
p = recnaux z S n -> suc (fst p) = suc (fst (recnaux z S n))
7 appeq2 
p = recnaux z S n -> S @ p = S @ recnaux z S n
8 6, 7 preqd 
p = recnaux z S n -> suc (fst p), S @ p = suc (fst (recnaux z S n)), S @ recnaux z S n
9 8 applame 
(\ p, suc (fst p), S @ p) @ recnaux z S n = suc (fst (recnaux z S n)), S @ recnaux z S n
10 9 conv recnaux 
(\ p, suc (fst p), S @ p) @ rec (0, z) (\ p, suc (fst p), S @ p) n = suc (fst (recnaux z S n)), S @ recnaux z S n
11 4, 10 ax_mp 
recnaux z S (suc n) = suc (fst (recnaux z S n)), S @ recnaux z 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)