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Mirrors > Home > ILE Home > Th. List > df-frec | GIF version |
Description: Define a recursive
definition generator on ω (the class of finite
ordinals) with characteristic function 𝐹 and initial value 𝐼.
This rather amazing operation allows us to define, with compact direct
definitions, functions that are usually defined in textbooks only with
indirect self-referencing recursive definitions. A recursive definition
requires advanced metalogic to justify - in particular, eliminating a
recursive definition is very difficult and often not even shown in
textbooks. On the other hand, the elimination of a direct definition is
a matter of simple mechanical substitution. The price paid is the
daunting complexity of our frec operation
(especially when df-recs 6170
that it is built on is also eliminated). But once we get past this
hurdle, definitions that would otherwise be recursive become relatively
simple; see frec0g 6262 and frecsuc 6272.
Unlike with transfinite recursion, finite recurson can readily divide definitions and proofs into zero and successor cases, because even without excluded middle we have theorems such as nn0suc 4488. The analogous situation with transfinite recursion - being able to say that an ordinal is zero, successor, or limit - is enabled by excluded middle and thus is not available to us. For the characteristic functions which satisfy the conditions given at frecrdg 6273, this definition and df-irdg 6235 restricted to ω produce the same result. Note: We introduce frec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Mario Carneiro and Jim Kingdon, 10-Aug-2019.) |
Ref | Expression |
---|---|
df-frec | ⊢ frec(𝐹, 𝐼) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})) ↾ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cF | . . 3 class 𝐹 | |
2 | cI | . . 3 class 𝐼 | |
3 | 1, 2 | cfrec 6255 | . 2 class frec(𝐹, 𝐼) |
4 | vg | . . . . 5 setvar 𝑔 | |
5 | cvv 2660 | . . . . 5 class V | |
6 | 4 | cv 1315 | . . . . . . . . . . 11 class 𝑔 |
7 | 6 | cdm 4509 | . . . . . . . . . 10 class dom 𝑔 |
8 | vm | . . . . . . . . . . . 12 setvar 𝑚 | |
9 | 8 | cv 1315 | . . . . . . . . . . 11 class 𝑚 |
10 | 9 | csuc 4257 | . . . . . . . . . 10 class suc 𝑚 |
11 | 7, 10 | wceq 1316 | . . . . . . . . 9 wff dom 𝑔 = suc 𝑚 |
12 | vx | . . . . . . . . . . 11 setvar 𝑥 | |
13 | 12 | cv 1315 | . . . . . . . . . 10 class 𝑥 |
14 | 9, 6 | cfv 5093 | . . . . . . . . . . 11 class (𝑔‘𝑚) |
15 | 14, 1 | cfv 5093 | . . . . . . . . . 10 class (𝐹‘(𝑔‘𝑚)) |
16 | 13, 15 | wcel 1465 | . . . . . . . . 9 wff 𝑥 ∈ (𝐹‘(𝑔‘𝑚)) |
17 | 11, 16 | wa 103 | . . . . . . . 8 wff (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) |
18 | com 4474 | . . . . . . . 8 class ω | |
19 | 17, 8, 18 | wrex 2394 | . . . . . . 7 wff ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) |
20 | c0 3333 | . . . . . . . . 9 class ∅ | |
21 | 7, 20 | wceq 1316 | . . . . . . . 8 wff dom 𝑔 = ∅ |
22 | 13, 2 | wcel 1465 | . . . . . . . 8 wff 𝑥 ∈ 𝐼 |
23 | 21, 22 | wa 103 | . . . . . . 7 wff (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼) |
24 | 19, 23 | wo 682 | . . . . . 6 wff (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼)) |
25 | 24, 12 | cab 2103 | . . . . 5 class {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))} |
26 | 4, 5, 25 | cmpt 3959 | . . . 4 class (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))}) |
27 | 26 | crecs 6169 | . . 3 class recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})) |
28 | 27, 18 | cres 4511 | . 2 class (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})) ↾ ω) |
29 | 3, 28 | wceq 1316 | 1 wff frec(𝐹, 𝐼) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})) ↾ ω) |
Colors of variables: wff set class |
This definition is referenced by: freceq1 6257 freceq2 6258 frecex 6259 frecfun 6260 nffrec 6261 frec0g 6262 frecfnom 6266 freccllem 6267 frecfcllem 6269 frecsuclem 6271 |
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