Muestra final “Rincones de mi cuerpo”

We identify in what follows the classes of formulas for which the substitution of⊥by F is possible, as in Berger et al. (2002). We define these classes in the same style as in the study presented in (Ishihara, 2000). We point out that they are distinct classes of formulas, since (Ishihara, 2000) does not restrict the logic to be minimal. We will give a detailed comparison in Section 3.5. Definition 3.5. Let P 6= ⊥ range over prime L-formulas, and ⊥ be a unary predicate variable. Let D, G, R and I range over D,G,R_D and I_G, respectively, and D0 be a decidable formula from D.

The classesD,G,RD andIG are simultaneously generated by the clauses

P,∀xD, R, I →D ∈ D

⊥, I, R→G, D0 →G ∈ G ⊥,∀xR, G→R ∈ RD

P,∀_xI, D →I ∈ I_G

Lemma 3.5. We can make the following generalizations:

• G~ →R∈ R_D • I~→D∈ D • R~ →G∈ G

• D~0 →G∈ G • D~ →I ∈ IG

Proof. Follows by induction on the number of formulas in the premise from the following observations

• ifG~ →R∈ RD and G0 ∈ G thenG0 →(G~ →R)∈ RD.

• ifI~→D∈ D and I0 ∈ IG then I0 →(I~→D)∈ D.

• ifR~ →G∈ G and R0 ∈ R_D thenR0→(R~ →G)∈ G.

• ifD~0 →G∈ G anD00 ∈ Ddecidable then D00 →(D~0→G)∈ G. • ifD~ →I ∈ I_G and D0 ∈ D thenD0 →(D~ →I)∈ I_G.

Remark 3.4. Clearly, R_D ⊂ D and I_G ⊂ G.

Comparison with Berger et al. (2002) In order to bridge Definition 3.5 and the one of the definite and goal formulas from Berger et al. (2002), we recall the latter:

Definition 3.6. (a) If a formula “ends” with ⊥, it is said to be relevant

and otherwise it is called irrelevant. Thus, C is a relevant formula iff

C:=⊥ |B →C| ∀_xC

(b) LetP∗ range over atomic formulas (including⊥). The definiteformulas

D∗ and goal formulas G∗ are defined inductively

D∗ := P∗ |G∗ →D∗ if D∗ relevant or G∗ irrelevant

| ∀xD∗,

G∗ := P∗ |D∗ →G∗ if D∗ relevant or D∗ quantifier-free

| ∀_xG∗ if G∗ irrelevant,

Remark 3.5. In the definition of the goal formula it is required that the premise ofG∗ is quantifier-free. The purpose is to allow case distinction for these formulas and as seen in Lemma 3.4, it suffices to consider decidable formulas, such that the above definition can be extended to:

Proposition 3.1. Let D,G,RD and IG be the classes of formulas as given

by Definition 3.5 and the relevant, definite and goal formulas as in Defini- tion 3.6. Then:

1. D are the definite formulas 2. G are the goal formulas

3. RD are the relevant definiteformulas

4. I_G are the irrelevant goal formulas.

Proof. LetP range over primeL-formulas,D∈ D,G∈ G,R ∈ RD,I ∈ IG

and D0 be a decidable formula fromD. Let P∗ range over prime formulas, including also ⊥, and D∗,G∗ be as in Definition 3.6.

We prove the claims by simultaneous induction on the formulas. 1. “⊂” We show: P,∀_xD, R, I →D are the definite formulas.

Clearly,P is definite.

∀xD is definite, becauseD is definite by the induction hypothesis.

Let R ∈ R_D. If R is ⊥, it is also definite. If it is ∀_xR, then R is by Remark 3.4 a (relevant) definite formula, so ∀xR is also a definite

formula. If G→ R, then it is a definite formula, since by hypothesis

G is a goal formula andR a (relevant) definite formula.

I →Dis definite, because I is an irrelevant goal formula (by Remark 3.4) and Dis a definite formula.

“⊃” We show: P∗, G∗ → D∗ (with D∗ relevant or G∗ irrelevant), ∀xD∗ ∈ D.

Cases P∗ and ∀xD∗ are trivial.

Case G∗ → D∗ . If D∗ is a relevant (definite) formula, then by the induction hypothesis D∗ ∈ RD. Since by (IH) G∗ ∈ G, it follows by

Remark 3.4 that G∗→D∗ ∈ R_D ⊂ D.

If D∗ is not relevant, then G∗ must be irrelevant, so G∗ ∈ I_G. Since

D∗ ∈ D by (IH), it follows thatG∗ →D∗ ∈ D.

2. ⊥, I, R → G, D0 → G are the goal formulas. ⊥ is clearly both in G and a goal formula.

CaseI ∈ IG. Since by (IH) IG is the class of irrelevant goal formulas,

For →, we have:

“⊂” The caseR→G∈ Gcoincides by (IH) withD∗ →G∗,Drelevant, so this is a goal formula.

Case D0 → G. If D0 is decidable, then by Remark 3.5 D0 → G is a goal formula.

“⊃” When D∗ → G∗ is a goal formula, then D∗ is either relevant or quantifier-free definite formula. In the first situation, we either have

R →G, which is clearly inG, orR →I. For the latter, observe that by the (IH) and Remark 3.4, I ∈ IG ⊂ G, so R → I ∈ G. If D∗ is a

quantifier-free definite formula, we are in the case D0 →G∈ G. 3. ⊥,∀xR, G→R∈ RD are the relevant definite formulas.

By Remark 3.4, R_D ⊂ D and since all above formulas “end” in ⊥, they are relevant.

For the reverse, if G→R is a relevant definite formula, thenR must be relevant and by (IH) this implies that R ∈ R_D. Since Gis a goal formula, then by (IH) G∈ G. Thus, we are in the caseG→R ∈ R_D. 4. P,∀_xI, D → I ∈ I_G are the irrelevant goal formulas. By Remark 3.4

and (IH) IG⊂ G, so by 2. such formulas are goal.

Since P does not range over ⊥, the formulas in IG cannot “end” in

bottom, so they are irrelevant by Definition 3.6.