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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 6210
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 6302 and frecsuc 6312.
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 4526. 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 6313, this definition and df-irdg 6275 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 6295 | . 2 class frec(𝐹, 𝐼) |
4 | vg | . . . . 5 setvar 𝑔 | |
5 | cvv 2689 | . . . . 5 class V | |
6 | 4 | cv 1331 | . . . . . . . . . . 11 class 𝑔 |
7 | 6 | cdm 4547 | . . . . . . . . . 10 class dom 𝑔 |
8 | vm | . . . . . . . . . . . 12 setvar 𝑚 | |
9 | 8 | cv 1331 | . . . . . . . . . . 11 class 𝑚 |
10 | 9 | csuc 4295 | . . . . . . . . . 10 class suc 𝑚 |
11 | 7, 10 | wceq 1332 | . . . . . . . . 9 wff dom 𝑔 = suc 𝑚 |
12 | vx | . . . . . . . . . . 11 setvar 𝑥 | |
13 | 12 | cv 1331 | . . . . . . . . . 10 class 𝑥 |
14 | 9, 6 | cfv 5131 | . . . . . . . . . . 11 class (𝑔‘𝑚) |
15 | 14, 1 | cfv 5131 | . . . . . . . . . 10 class (𝐹‘(𝑔‘𝑚)) |
16 | 13, 15 | wcel 1481 | . . . . . . . . 9 wff 𝑥 ∈ (𝐹‘(𝑔‘𝑚)) |
17 | 11, 16 | wa 103 | . . . . . . . 8 wff (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) |
18 | com 4512 | . . . . . . . 8 class ω | |
19 | 17, 8, 18 | wrex 2418 | . . . . . . 7 wff ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) |
20 | c0 3368 | . . . . . . . . 9 class ∅ | |
21 | 7, 20 | wceq 1332 | . . . . . . . 8 wff dom 𝑔 = ∅ |
22 | 13, 2 | wcel 1481 | . . . . . . . 8 wff 𝑥 ∈ 𝐼 |
23 | 21, 22 | wa 103 | . . . . . . 7 wff (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼) |
24 | 19, 23 | wo 698 | . . . . . 6 wff (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼)) |
25 | 24, 12 | cab 2126 | . . . . 5 class {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))} |
26 | 4, 5, 25 | cmpt 3997 | . . . 4 class (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))}) |
27 | 26 | crecs 6209 | . . 3 class recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})) |
28 | 27, 18 | cres 4549 | . 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 1332 | 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 6297 freceq2 6298 frecex 6299 frecfun 6300 nffrec 6301 frec0g 6302 frecfnom 6306 freccllem 6307 frecfcllem 6309 frecsuclem 6311 |
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