Translating Between Hypersequents and Labelled Sequents
7.2. TRANSLATING HYPERSEQUENTS TO LABELLED SEQUENTS
handled by extending the language of simply labelled sequents with a placeholder for the empty antecedents or succedents and updating the definitions of translations, as well as equivalence modulo permutation. This is discussed in Appendix E on page 263.
Theorem 7.3 (Truth Preservation). Let I be an interpretation, and let H ∈ Seq+such that I H. Then for all non-empty lists ¯x ∈ Lab+0, I sls3 ¯x H .
Proof. There exists the kth component Sk∈ H , where 1 ≤ k ≤ | ¯x|, such that I Sk(Defi-
nition 4.10 on page 70). Let xk be the kth label in a list ¯x of labels. Let Sxkk = sls2 xkSk.
I Sxk
k . Let S = sls3 ¯x H . Since there exists a label xk∈ lab(S ) such that I S xk, I S (Definition 5.31 on page 109). So I sls3 ¯x H .
Corollary 7.4 (Countermodel Preservation). Let I be an interpretation, and let H be a hypersequent such that I 2 H. Then for all non-empty lists ¯x ∈ Lab+0, I 2 sls3 ¯x H .
Proof. The proof follows similarly to Theorem 7.3.
The translation function can be extended to cover schematic hypersequents by adding cases for metavariables. We first define the sets of metavariables needed for the translation between schematic hypersequents and schematic simply labelled sequents.
Recall the definition of the set (Seqµ+ HyperseqVar)+ of schematic hypersequents given in Definition 4.15 on page 71.
Definition 7.5. Let slsvar be a function from HyperseqVar to a unique multil- abelled sequent in MultisetVar, e.g. from H to Γ†⇒∆†. (The † subscript is used to indicate that these metavariables are in the denumerable set MultisetVar†, and distinct
from metavariables used to indicate unilabelled multisets.)
Proposition 7.6. slsvar is a 1 − 1 function.
154 7. TRANSLATING BETWEEN HYPERSEQUENTS AND LABELLED SEQUENTS
Definition 7.7. We define a translation slsµ3of schematic solid hypersequents to sim- ply labelled sequents below:
slsµ0 xα = αx where α ∈ Prop+ MultisetVar slsµ1 xΓ = (slsµ0 x) ~ Γ whereΓ ∈ (Prop + MultisetVar)∗ where x ∈ Lab0. slsµ2 x S =
(slsµ1 x) ~ S where S ∈ (Prop+ MultisetVar)∗2 (slsvar S ) where S ∈ HyperseqVar
where x ∈ Lab0. slsµ3 ¯x (α1| ... | αn)= slsµ2 x1α1 if n= 1 (slsµ3 ¯x (α1| ... | αn−1)) t (slsµ2 xnαn) otherwise
where ¯x= x1,..., xn∈ Lab+0 is a list of labels.
Remark 7.8. Note that the set Prop of formulae is identical to the set of metaformu- lae. The difference is one of interpretation: a propositional variable in a metaformula may represent an arbitrary formula rather than an atomic proposition.
Likewise schematic sequents may be treated as sequents where some of the formulae are metavariables that denote arbitrary multisets of formulae.
Example. The following schematic hypersequent
H | H0|Γ1⇒∆1|Γ2⇒∆2, A ∨ B
would be translated by the slsµ3function and a list of labels x1,..., x4into the schematic
simply labelled sequent
Γ†,Γ 0 †Γ x3 1 ,Γ x4 2 ⇒∆†,∆ 0 †,∆ x3 1 ,∆ x4 2 ,(A ∨ B) x4
modulo the relationship between hypercontext variables and simply labelled sequent vari- ables in slsvar.
7.2. TRANSLATING HYPERSEQUENTS TO LABELLED SEQUENTS 155
Corollary 7.9 (Translation of Rules). Let ρ be a primitive n-premiss hypersequent rule,
H1 ... Hn H0
ρ
in a calculus HGS for a logic S. Thenρ can be translated into a corresponding simply labelled sequent rule
(slsµ3 M1H1) ... (slsµ3 MnHn)
(slsµ3 M0H0)
ρ
where all M0,..., Mnare mutually disjoint, for the corresponding simply labelled calculus
LGS for a logic S.
Proof. Follows from Theorem 7.3 on page 153, when ρ is instantiated. Remark 7.10. The use of disjoint lists of labels for each premiss and the conclusion is acceptable because of the semantics of hypersequents and simply labelled sequents: the actual label names play no rôle in the inference.
Corollary 7.11. Let ρ be a (depth-preserving) admissible n-premiss hypersequent rule,
H1 ... Hn H0 ρ
in a calculus HGS for a logic S. Thenρ can be translated into a corresponding (depth- preserving) admissible rule
(slsµ3 M1H1) ... (slsµ3 MnHn)
(slsµ3 M0H0)
ρ
where all M0,..., Mnare mutually disjoint, for the corresponding simply labelled calculus
LGS for a logic S.
Proof. Given a proof of admissibility of ρ in HGS, one can give a corresponding proof of admissibility of ρ in LGS by translating all rules in all (sub)cases appropriately. Remark 7.12. Corollary 7.9 applies to admissible as well as primitive rules. One can translate admissibility proofs for a hypersequent calculus to admissibility proofs for the translated simply labelled sequent calculus by translating the rules in each case.
156 7. TRANSLATING BETWEEN HYPERSEQUENTS AND LABELLED SEQUENTS
Remark 7.13. One can use the following “rules of thumb” for preserving labels be- tween premisses and the conclusion without affecting soundness:
(1) Internal hypersequent rules will be translated to labelled sequent rules with only one explicit label in each premiss and the conclusion, so clearly these should all be the same label.
(2) The trivially invertible forms of hypersequent rules, or rules where a component is unchanged in a premiss and the conclusion, should be translated to that corre- sponding slices are equivalent modulo permutation of labels. Such slices should have the same label between premiss(es) and conclusion. However, multiple slices in the same premiss that are equivalent modulo permutation should not have the same label (e.g. a translation of the EC rule), as each active component corresponds to a different label.
(3) If the antecedents (resp. succedents) of a slice in a premiss (or premisses) and another slice in the conclusion are equivalent modulo permutation, then they can share a label.
Preserving labels between the premisses and conclusion will be needed for translating simply labelled rules into relational rules in Chapter 9.
Proposition 7.14. The rules given in Remark 7.13 preserve soundness.
Proof. By the disjunctive semantics of labelled sequents.
Example. The following hypersequent rule
H |Γ⇒ A⊃ B,∆ | Γ, A⇒ B H |Γ⇒ A⊃ B,∆ R ⊃ι can translated to the simply labelled sequent rule
Γ0,Γx,Γy, Ay⇒∆0,(A⊃ B)x,∆x, By
Γ0,Γx⇒∆0,(A⊃ B)x,∆x R ⊃ι
where x,y#Γ0,∆0. (The † notation is omitted for simplicity, since it is clear that there is no overlap in names for the variables.) The x label is shared because the slice in the premiss and conclusion are equivalent modulo permutation.
7.2. TRANSLATING HYPERSEQUENTS TO LABELLED SEQUENTS 157
To simplify the notation, the restriction that x#∆0can be relaxed and∆xabsorbed into ∆0
, since it does not interact with any other variable in the rule.
Example. The following hypersequent rule H |Γ⇒∆ | Γ, A⇒ B
H |Γ⇒ A⊃ B,∆ R ⊃
0
can translated to the simply labelled sequent rule Γ0,Γx,Γy, Ay⇒∆0,∆x, By
Γ0,Γx⇒∆0,(A⊃ B)x,∆x R ⊃0
where x,y#Γ0,∆0. The x label is shared because the slice in the premiss and conclusion have antecedents that are equivalent modulo permutation.
7.2.1. The framework LG3ipm∗. The system LG3ipm (Figure 7.1) can be obtained from the hypersequent calculus HG3ipm (Figure 4.4 on page 93) using the above proce- dure. LG3ipm∗ consists of rules of LG3ipm plus the rules from Figure 7.2 on page 159, which are obtained from the corresponding hypersequent rules.
Px,Γ⇒∆, Px Ax ⊥x,Γ⇒∆ L⊥ Γ, Ax, Bx⇒∆ Γ, A ∧ Bx⇒∆ L∧ Γ⇒ Ax,∆ Γ⇒ Bx,∆ Γ⇒ A ∧ Bx,∆ R∧ Γ, Ax⇒∆ Γ, Bx⇒∆ Γ, A ∨ Bx⇒∆ L∨ Γ⇒ Ax, Bx,∆ Γ⇒ A ∨ Bx,∆ R∨ (A ⊃ B)x,Γ⇒∆, Ax Bx,Γ⇒∆ (A ⊃ B)x,Γ⇒∆ L ⊃ Ax,Γ0⇒∆0, Bx Γ0⇒∆0,∆x,(A⊃ B)x R ⊃ where x#∆0in R ⊃. Γ0⇒∆0 Γx,Γ0⇒∆0,∆x EW Γx,Γy,Γ0⇒∆0,∆x,∆y Γx,Γ0⇒∆0,∆x EC
where x, y#Γ0,∆0in EW and EC.
Figure 7.1. The simply labelled calculus LG3ipm.
158 7. TRANSLATING BETWEEN HYPERSEQUENTS AND LABELLED SEQUENTS
Proof. Soundness follows from the soundness and completeness of HG3ipm. Proposition 7.16. The structural rules
Γ⇒∆ Γ, Ax⇒∆ (LW) Γ⇒∆ Γ⇒ Ax,∆ (RW) Γ, Ax, Ax⇒∆ Γ, Ax⇒∆ (LC) Γ⇒ Ax, Ax,∆ Γ⇒ Ax,∆ (RC)
are depth-preserving admissible in LG3ipm.
Proof. Straightforward. Note that for weakening rules, we do not need to assume that the principal label occurs in the sequent.
Proposition 7.17. The logical rules of LG3ipm are depth-preserving invertible.
Proof. Semantically, by translation from HG3ipm. Proposition 7.18 (Label substitution). The following label substitution rule
S
[y/x]S [y/x]
is depth-preserving admissible in LG3ipm.
Proof. By induction on the derivation depth. Lemma 7.19. The general weakening and general contraction rules
Γ0⇒∆0
Γx,Γ0⇒∆0,∆x (GW)
Γy,Γx,Γ0⇒∆0,∆x,∆y
Γx,[y/x]Γ0⇒ [y/x]∆0,∆x (GC)
where are depth-preserving admissible in LG3ipm.
Proof. GW from multiple instances of LW and RW. GC from substitution (Proposi- tion 7.18), then multiple instances of LC and RC. Remark 7.20. These rules correspond to the GW and GC hypersequent rules (Lem- mas 4.32 and 4.33 on page 74).
Corollary 7.21 (External Rules). Instances of EW and EC external weakening and external contraction rules
Γ0⇒∆0
Γx,Γ0⇒∆0,∆x (EW)
Γy,Γx,Γ0⇒∆0,∆x,∆y
Γx,Γ0⇒∆0,∆x (EC)
7.3. TRANSLATING LABELLED SEQUENTS TO HYPERSEQUENTS 159