ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-frec GIF version

Definition df-frec 6254
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 6168 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 6260 and frecsuc 6270.

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 4486. 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 6271, this definition and df-irdg 6233 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.)

Assertion
Ref Expression
df-frec frec(𝐹, 𝐼) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))})) ↾ ω)
Distinct variable groups:   𝑥,𝑔,𝑚,𝐹   𝑥,𝐼,𝑔,𝑚

Detailed syntax breakdown of Definition df-frec
StepHypRef Expression
1 cF . . 3 class 𝐹
2 cI . . 3 class 𝐼
31, 2cfrec 6253 . 2 class frec(𝐹, 𝐼)
4 vg . . . . 5 setvar 𝑔
5 cvv 2658 . . . . 5 class V
64cv 1313 . . . . . . . . . . 11 class 𝑔
76cdm 4507 . . . . . . . . . 10 class dom 𝑔
8 vm . . . . . . . . . . . 12 setvar 𝑚
98cv 1313 . . . . . . . . . . 11 class 𝑚
109csuc 4255 . . . . . . . . . 10 class suc 𝑚
117, 10wceq 1314 . . . . . . . . 9 wff dom 𝑔 = suc 𝑚
12 vx . . . . . . . . . . 11 setvar 𝑥
1312cv 1313 . . . . . . . . . 10 class 𝑥
149, 6cfv 5091 . . . . . . . . . . 11 class (𝑔𝑚)
1514, 1cfv 5091 . . . . . . . . . 10 class (𝐹‘(𝑔𝑚))
1613, 15wcel 1463 . . . . . . . . 9 wff 𝑥 ∈ (𝐹‘(𝑔𝑚))
1711, 16wa 103 . . . . . . . 8 wff (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚)))
18 com 4472 . . . . . . . 8 class ω
1917, 8, 18wrex 2392 . . . . . . 7 wff 𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚)))
20 c0 3331 . . . . . . . . 9 class
217, 20wceq 1314 . . . . . . . 8 wff dom 𝑔 = ∅
2213, 2wcel 1463 . . . . . . . 8 wff 𝑥𝐼
2321, 22wa 103 . . . . . . 7 wff (dom 𝑔 = ∅ ∧ 𝑥𝐼)
2419, 23wo 680 . . . . . 6 wff (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))
2524, 12cab 2101 . . . . 5 class {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))}
264, 5, 25cmpt 3957 . . . 4 class (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))})
2726crecs 6167 . . 3 class recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))}))
2827, 18cres 4509 . 2 class (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))})) ↾ ω)
293, 28wceq 1314 1 wff frec(𝐹, 𝐼) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))})) ↾ ω)
Colors of variables: wff set class
This definition is referenced by:  freceq1  6255  freceq2  6256  frecex  6257  frecfun  6258  nffrec  6259  frec0g  6260  frecfnom  6264  freccllem  6265  frecfcllem  6267  frecsuclem  6269
  Copyright terms: Public domain W3C validator