Theorem rlrecS ≪ | index | src | ≫ 

theorem rlrecS (S: set) (a l z: nat):
  $ rlrec z S (l |> a) = S @ (l, a, rlrec z S l) $;
StepHypRefExpression
1 eqtr 
rlrec z S (l |> a) = lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (a : rev l) ->
  lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (a : rev l) = S @ (l, a, rlrec z S l) ->
  rlrec z S (l |> a) = S @ (l, a, rlrec z S l)
2 lreceq3 
rev (l |> a) = a : rev l -> lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (rev (l |> a)) = lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (a : rev l)
3 2 conv rlrec 
rev (l |> a) = a : rev l -> rlrec z S (l |> a) = lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (a : rev l)
4 revsnoc 
rev (l |> a) = a : rev l
5 3, 4 ax_mp 
rlrec z S (l |> a) = lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (a : rev l)
6 1, 5 ax_mp 
lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (a : rev l) = S @ (l, a, rlrec z S l) -> rlrec z S (l |> a) = S @ (l, a, rlrec z S l)
7 eqtr 
lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (a : rev l) = (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) @ (a, rev l, rlrec z S l) ->
  (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) @ (a, rev l, rlrec z S l) = S @ (l, a, rlrec z S l) ->
  lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (a : rev l) = S @ (l, a, rlrec z S l)
8 lrecS 
lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (a : rev l) =
  (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) @ (a, rev l, lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (rev l))
9 8 conv rlrec 
lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (a : rev l) = (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) @ (a, rev l, rlrec z S l)
10 7, 9 ax_mp 
(\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) @ (a, rev l, rlrec z S l) = S @ (l, a, rlrec z S l) ->
  lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (a : rev l) = S @ (l, a, rlrec z S l)
11 revrev 
rev (rev l) = l
12 anlr 
a1 = a /\ a2 = rev l /\ a3 = rlrec z S l -> a2 = rev l
13 12 reveqd 
a1 = a /\ a2 = rev l /\ a3 = rlrec z S l -> rev a2 = rev (rev l)
14 11, 13 syl6eq 
a1 = a /\ a2 = rev l /\ a3 = rlrec z S l -> rev a2 = l
15 anll 
a1 = a /\ a2 = rev l /\ a3 = rlrec z S l -> a1 = a
16 anr 
a1 = a /\ a2 = rev l /\ a3 = rlrec z S l -> a3 = rlrec z S l
17 15, 16 preqd 
a1 = a /\ a2 = rev l /\ a3 = rlrec z S l -> a1, a3 = a, rlrec z S l
18 14, 17 preqd 
a1 = a /\ a2 = rev l /\ a3 = rlrec z S l -> rev a2, a1, a3 = l, a, rlrec z S l
19 18 appeq2d 
a1 = a /\ a2 = rev l /\ a3 = rlrec z S l -> S @ (rev a2, a1, a3) = S @ (l, a, rlrec z S l)
20 19 applamed 
a1 = a /\ a2 = rev l -> (\ a3, S @ (rev a2, a1, a3)) @ rlrec z S l = S @ (l, a, rlrec z S l)
21 20 appslamed 
a1 = a -> (\\ a2, \ a3, S @ (rev a2, a1, a3)) @ (rev l, rlrec z S l) = S @ (l, a, rlrec z S l)
22 21 appslame 
(\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) @ (a, rev l, rlrec z S l) = S @ (l, a, rlrec z S l)
23 10, 22 ax_mp 
lrec z (\\ a1, \\ a2, \ a3, S @ (rev a2, a1, a3)) (a : rev l) = S @ (l, a, rlrec z S l)
24 6, 23 ax_mp 
rlrec z S (l |> a) = S @ (l, a, rlrec z S 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)