Coming to terms with modal logic : on the interpretation of modalities in typed lambda-calculus

ion G 1A: Bis G 1(>.x : C.d) : (ITx : C.D), an immediate consequence of (1) G, x : C 1d : D and (2) G 1(IIx: C.D) : s. By lH (on (1)) FV(d),FV(D) ç FV(G,x: C) and (on (2)) FV(ITx: C.D),FV(s) ç FV(G,x: C). Hence FV( C) Ç FV( G) and FV(Àx : C.d) Ç FV( G) since by definition FV(Àx : C.d) = FV(C) u (FV(d){x}) (FV(ITx : C.D) = FV(C) u (FV(D){x})). Notice that by Free Variabie Lemma (i) and (1), x (/: FV(G). Therefore FV(>.x : C.d),FV(ITx: C.D) Ç FV(G). The transfer, import and export-cases again use the Free Variabie definition FV( G IQ] t:) = FV( G). Proof of (iii). The cases forstart and weakening use the part (ii) of the Free Variabie Lemma, we do the case for start: Start G 1A: Bis G', x : C 1x : Ca direct consequence of G' 1C : s. By lH for all A; in G', FV(A;) Ç {x1, ... , x;_t}. We have to prove that FV( C) Ç { x1, ... , xi} = FV( G'). But si nee G' 1C : s, we have FV ( C) Ç F V ( G') by the previous clause (i i) of the Free Variabie Lemma, hence FV(Ai) Ç {x1, ... , x;_ I} for all A; in G. For the transfer, import and export-cases we need the simple observation that A; in G are the A; in G IQl E 3.2.3. LEMMA. Start Lemma Let G be a legal generalized context. Then {i) IJ s1 : s2 is a typing axiom {E A Ty1>e ), G 1s1 : s2 3.2. PRELIMINARIES 97 {ii} IJ c : A is a logica/ axiom, (E ALogi<: ), G f-c : A : Prop {iii} Ij (x : A) E r in G' tQJ r {where G ::: G' IQ! r }, then G' IQ! r f-x : A PROOF. Proof of (i),(ii) and (iii) . By assumption of G f-A: B forsome A: B. The result follows by induction on the derivation of G 1A : B. Proof of (i). The cases for transfer and import all use the transfeTJ.-rule: Transfer2 G f-A: B is G' tQJ t: fC : D where G = G'IQJ t:, an immediate consequence of G' fC : D : Type. By IH G' f-SI : s2, but then by transfer1 G' IQJ E fSJ : s2. Hence G fSJ : s2, for SJ : s2 E A Type . For the export-cases we have look further back in the derivation in applying the IH, we show the case for K -export: K -export G f-A : B is G f-kC : DD an immediate consequence of G tQJ E fC : D : Prop. Applying the IH to the last step in the derivation does not work bere, however since all derivations start from the context t:; e we can go up in the derivation tree to find the place where the IQ) was introduced going from G to G IQJ e for the first time. This means sarnething must have been derivable on G before, and since this derivation is shorter, IH gives that G f-SJ : s2, for SJ : s2 E A Type. The proof of (ii) is completely analogous. Proof of (iii). The proof is straightforward for non-modal cases, and trivia! for the transfer and import-rules: Transfer1 G fA : B is G'IQJ t: f-C : s where G = G'IQJ e, animmedia te consequence of G' f-C : s. Note that this cannot occur when G is 'non-blocked' ( G = r). Therefore we treat the case of the 'complex' context G' IQJ e. Since r = e, it contains no variables, and so trivially G f-x : C if (x : C) Er. For the export-cases we use an argument similar to the one given above in the proof of (i). Given the following definitions of substitution on, and concatenation of generalized contexts, a Substition Lemma can be proved: 3.2.4. DEFINITION. Subtitution, concatenation On a generalized context 1::. = !::.1 ~ ... IQI l::.n, the substitution of a term D fora variabie x yields !::.[x :=DJ = t:.t[x := D]IQJ . .. IQ! l::.n[x :=Dj. Given two generalized contexts G ::: f 1 IQJ . .. IQJ r m and t:. = 1::.1 IQJ ••• IQJ l::.n, their concatenation G, t:. = f1 IQJ ••• IQJ r m, !::.1 IQJ ••• IQJ l::.n . 3.2.5. LEMMA. Substitution Lemma Assume (1) G, x : C, 1::. 1A : B and (2) G 1D : C, where G and 1::. are generalized pseudo-contexts. Then G, !::.[x:= Dj f-A[x := D]: B[x :=DJ . 98 CHAPTER 3. META THEORY OF MPTSs PROOF. Proof. By induction on (the lengthof the) derivation of (1), where M* is used as an abbreviation for M[x := DJ. The non-modal cases are analogous to those in the proof of the Substitution Lemma for PTS's. The modal cases require some calculations with the definitions for substitution, e.g. : K-export G,x: C,/1 1A : Bis G,x: C, 1kE: OF an immediate consequence of G,x: C,/1 li:lle 1E: F: Prop. By lH G,(/1 li:lle)* 1E*: F*(: Prop). By definition (!1 fQJ e)* = !1* 101 e* = !1* 101 E, since E* = E. Hence G, (!1 [QJ E)> = G, f1• [QJ E, and so G, !1* fQJ E 1E*: F*(: Prop). Therefore by K-export G, !1* 1kE*: D(F*), and since (kE*) := (kE)* (FV(kE) = FV(E)) and D(F*) := (DF)*, G, !1* 1(kE)*: (DF)*. The proof of a Thinning Lemma is not completely straightforward. The transfer and import-rules are formulated in such a way that they yield a new generalized context of the iorm G fQJ e. However, to prove Thinning we have to show that there are derived versionsof these rules that yield generalized contexts G fQJ r, for an arbitrary 'non-blocked' context r. To prove this, the following lemma is needed: 3.2.6. LEMMA. Legality Lemma Ij G fQJ f 1 , x : C is leg al then G lQl f 1 r C : s. PROOF. By induction on the length of the derivation of G fQJ T1, x : C 1A : B . Except for the axiom cases which cannot occur (since G fQJ f 1, x : C t;. e) and start and weakening which are immediate, the non-madal cases are regular. The transfer and import-cases cannot occur: Transfer1 G 19 f 1, x : C 1A : B is G IQJ E 1D : s , an immediate consequence of G 1D : s. This case cannot occur: G IQJ f 1, x : C '1. G fQJ e. and the export-cases require some additional reasoning: K -export G fQJ f 1, x : C 1A : B is G 1kD : DE an immediate consequence of G 101 f 1 , x : c 101 E: 1D : E : Prop. Since all derivations start from E and are finite, we can go up in the tree to find the place where the 101 was introduced, going from G IQI f 1 , x : C, to G IJ f 1, x : C 101 e for the first time. This means that sarnething must have been derivable on G ~ f 1 , x : C before, and since this denvation is shorter IH gives us that G lQl r-' 1C : s. 3.2.7. LEMMA. Derived Rules Lemma The following are derived rules in an MPTS: 1 GI-A :s Transfer1 G r A IQI 1: s Tra , 1 G 1A : B : Set ns,er3 G 101 r 1A : B K . 1 G 1A : DB: Prop tmport _ · G IQI r 1kA: B . I G r A: -.OB: Prop 5 tmport • G lQl r 1SA : -.DB T 11 GI-A:B:Type rans,er2 G !QJ r 1A : B Th :j 1 G 1c : A : Prop ans er ax G fQJ r 1c : A 1 G 1A : DB : Prop 4 import • G 101 r 14A : DB B . 1 G 1A : B : Prop tmport G !QJ r 1bA : -.o-.B where r is a {non-blocked) pseudocontext such that G IQ! r is legal. 3.2. PRELIMINARIES 99 Given the original rules of Transfer and Import, proving the following is sufficient: 1 IfGIQJe 1A : s then G !dl r 1A : s . 2 IfGIQJe 1A : B(: Type) then G IQJ r 1A: B. 3 IfGIQJe 1A: B(: Set) then G l!:ll r 1A: B. 4 IfGIQJe 1c: A(: Prop) then G IQJ r 1c: A. 5 IfGIQJe 1kA : B then G IQJ r 1kA : B 6 IfGIQJe 14A : DB then G !bil r 14A : DB 7 lfGIQJe 1SA : -.OB then G IQJ r 1SA : -.OB 8 IfGIQJe 1bA : -,0-,B then G IQJ r 1bA : -,0-,B PROOF. By induction on the lengthof f . The basic case for r =: e is immediate by the above. The induction case where r =: f', x : C is the same for all cases, we show 1: 1 By lH G IQI r' 1A : s, and by the Legality Lemma ( G IQJ r' is legal) G IQJ r' 1C : s, hence weakening yields G IQI f', x : C 1A : s and G IQJ r fA: s. Now we can prove a Thinning Lemma for the modal systems, using the 'subset relation' for generalized contexts defined earlier. 3.2.8. LEMMA. Thinning Lemma Let G and /::;. be leg al generalized pseudocontexts such that G Ç /::;.. Th en if G 1A : B, ó. 1A: B. PROOF. By induction on the lengthof the derivation of G 1A : B. The cases for transfer and import require the derived forms of the these rules from the derived rules lemma. We show the case for K -import: K-import G 1A : B is G' !bil e 1kc : D where G = G' IQI e, an immediate consequence of G' 1C : DD : Prop. Since G = G' IQJ e and G Ç ó., it must be the case that ó. = ö.' IQJ r forsome r (since for all r, e Ç r) and G' Ç ó.'. Hence by lH ö.' 1C: DD: Prop , so by the derived rule K-import ' I::;.' IQ! r 1kC: D and therefore ~::;. 1kc: n. In the Export-cases we have to show that /::;. is legal before the IH can be applied: K -export G 1A : B is G 1kC : DD an immedia te consequence of G lbll e 1C : D : Prop. Since G Ç /::;. and e Ç e, by definition G IQJ e Ç /::;.IQJ e. Furthermore ó.lbll eis !ega!: /::;. is legal, hence by the Start Lemma (i) /::;. fs1 : s2 for s1 : s2 E A Type (note that A Type f= 0), and by trans j eT}, /::;. IQJ e 1s1 : s2. Therefore by lH /::;. IQJ e 1C : D : Prop, andsoó. 1kC:DD . 100 CHAPTER 3. META THEORY OF MPTSs 3.2.9. COROLLARY. Strong Thinning For terms {A) that are not proofs {nat A : B : Prop), we can prove a stronger result (jor cases where G is an intialpart of t:.) by combining Thinning with the transfer rule. Let G and t:. be /ega/ generalized pseudocontexts such that G ~ t:.. Th en (i) ij G 1A : sI t:. 1A : s. {ii) ij G 1A : B : Type, t:. 1A : B : Type. {iii) ij G 1A : B : Set, t:. 1A : B :Set. PROOF. Proof of (i), (ii), and (iii). By construction of t:. from G while preserving the derivability of A : s . We do the proof of (ii): Suppose that G 1A : B : Type for some terms A and B: (1) f1 IQ! ••• ~ r m 1A : B : Type Since 'v'i(1 ~i ~ m)(f; Ç r~), we can conclude by Thinning (2) r~ !':ll ••. !':ll r~ 1A : B : Type Now the transfer2 rule can be applied to obtain r~ IQJ • • • l!:ll r~ IQI f 1A : B and by transfer1 we have r~~CJ ... Ii:!lr~~f 1B:Type