Deduction modulo and Supernatural deduction are two extentions of predicate logic with computation rules. Whereas the application of computation rules in deduction modulo is transparent, these rules are used to build non-logical deduction rules in Supernatural deduction. In both cases, adding computation rules may jeopardize proof normalization, but various conditions have been given in both cases, so that normalization is preserved. We prove in this paper that normalization in Supernatural deduction and in Deduction modulo are equivalent, i.e. the set of computation rules for which one system strongly normalizes is the same as the set of computation rules for which the other is

Brauner, Paul

Dowek, Gilles

Wack, Benjamin

HAL-Polytechnique

Normalization in Supernatural deduction and inDeduction moduloPaul Brauner, Gilles Dowek, Benjamin WackTo cite this version:Paul Brauner, Gilles Dowek, Benjamin Wack. Normalization in Supernatural deduction and inDeduction modulo. 2007. <inria-00141720v3>HAL Id: inria-00141720https://hal.inria.fr/inria-00141720v3Submitted on 16 Nov 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Normalization in Supernatural deduction and inDeduction moduloPaul Brauner1, Gilles Dowek2, Benjamin Wack31 E´cole polytechnique and INRIA,LIX, E´cole polytechnique, 91128 Palaiseau Cedex, Francegilles.dowek@polytechnique.edu http://www.lix.polytechnique.fr/~dowek/2 Loria, Campus Scientifique - BP 239 - 54506 Vandoeuvre-le`s-Nancy Cedex, Francepaul.brauner@loria.frhttp://www.loria.fr/~brauner/3 Loria, Campus Scientifique - BP 239 - 54506 Vandoeuvre-le`s-Nancy Cedex, FranceAbstract. Deduction modulo and Supernatural deduction are two ex-tentions of predicate logic with computation rules. Whereas the applica-tion of computation rules in deduction modulo is transparent, these rulesare used to build non-logical deduction rules in Supernatural deduction.In both cases, adding computation rules may jeopardize proof normal-ization, but various conditions have been given in both cases, so thatnormalization is preserved. We prove in this paper that normalizationin Supernatural deduction and in Deduction modulo are equivalent, i.e.the set of computation rules for which one system strongly normalizes isthe same as the set of computation rules for which the other is.1 IntroductionIntegrating a theory into a deduction system has several advantages over workingwith axioms. It allows proof-search algorithms to avoid some redundancies, itpermits in some cases to get witnesses from constructive proofs and hence toprogram with proofs, etc.One way to do so is to formulate the theory of interest as a rewrite system andidentify congruent formulae. This is the idea of deduction modulo. For instance,the axiom ∀X∀Y ((X ⊆ Y ) ⇔ ∀z (z ∈ X ⇒ z ∈ Y )) may be formulated as therewrite rule (X ⊆ Y )→ ∀z (z ∈ X ⇒ z ∈ Y ) and(∀I) Γ ` x ∈ A⇒ x ∈ BΓ ` A ⊆ Bis then a valid deduction step. Thus the natural deduction rules have to bereformulated in order to take the congruence into account.An more deterministic alternative to deduction modulo is Supernatural De-duction where the rewrite rules are translated into new deduction rules in thelogic. For instance, the rewrite rule (X ⊆ Y )→ ∀z (z ∈ X ⇒ z ∈ Y ) yields thefollowing introduction and elimination rules∀-intro Γ ` (z ∈ X ⇒ z ∈ Y )Γ ` X ⊆ Y z /∈ FV(Γ ) ∀-elimΓ ` X ⊆ YΓ ` t ∈ X ⇒ t ∈ Ywhere all connectors and quantifiers have disappeared. It may be argued thatthese rules are much closer to usual mathematical practice than either predicatelogic with axioms or Deduction Modulo.Since soundness and completeness with respect to predicate logic with axiomshas been proved for both systems, they prove the same theorems but we areinterested in a stronger equivalence property.To explain this strong equivalence property, we must first notice that in bothcases, proof normalization and cut elimination are jeopardized. However, variousconditions have been given so that normalization is preserved. For deductionmodulo, such criteria have been developed in [1,2,3]. For SND, criteria havebeen developed in [4].This raises the following problem: are there some rewrite system for whichone deduction system has the proof normalization property but not the other ordoes proof normalization depend only on the rewrite system itself ? We provein this paper that this is indeed the case: given an orthogonal rewrite system Rrewriting atomic formulae to arbitrary formulae, proofs normalize in DeductionModulo(R) if and only if they normalize in Supernatural Deduction(R). As aconsequence, the semantic criteria developed for Deduction Modulo [3] may betransferred to supernatural deduction.Our result is limited to minimal logic (i.e. our connectives are restricted toimplication and universal quantification). The result can be extended to con-jonction but the extension to the disjunction, the existential quantifier and con-tradiction is more challenging. Indeed, Supernatural Deduction rules are alreadytrickier to define when such connectors and quantifiers are involved. A possiblesolution may be to move from natural deduction to sequent calculus [5].To prove this equivalence between Supernatural Deduction and DeductionModulo, we shall prove that they are both equivalent to a third and older systemdue to Prawitz: Natural Deduction with folding and unfolding rules. Actually, theequivalence of Deduction modulo and Prawitz’ system has already been provedin [6]. Thus, all that remains to be done is to prove the equivalence betweenPrawitz’ system and Supernatural Deduction. Fortunately, this can be achievedby a simple syntactic translation of proofs.This paper contributes to show that the choice of a formalism or anotherseems to be a rather superficial choice, as properties such as strong normalizationare quite stable when one switches from one system to another.2 From fold/unfold to supernatural deduction2.1 Natural deductionOur starting point is natural deduction for minimal logic with five rules: (Ax),(⇒I), (⇒E), (∀I), (∀E). We use a lambda calculus notation for proofs:T ::= α | λα : A.T | T T | λx.T | T tThe deduction rules enriched with proof-terms are the following: but these rulesare typed with different typing rules(Ax)Γ, α : A ` α : A(⇒I) Γ, α : A ` pi : BΓ ` (λα : A.pi) : A⇒ B (⇒E)Γ ` pi1 : A⇒ B Γ ` pi2 : AΓ ` (pi1 pi2) : B(∀I) Γ ` pi : AΓ ` (λx.pi) : ∀xA x /∈ FV(Γ ) (∀E)Γ ` pi : ∀xAΓ ` (pi t) : A{x := t}The reduction rules on these proof-terms are the two kinds of β-reduction((λα : A pi1) pi2)B pi1{α := pi2}((λx pi) t)B pi{x := t}2.2 Natural deduction with folding and unfolding rulesConsider an orthogonal rewrite system R rewriting atomic formulae to formulae.For a rewrite rule P → ϕ, we may add Prawitz’ folding and unfolding rules[7] replacing ϕ by P and P by ϕ. To do so, we enrich the proof-terms language.T ::= α | λα : A.T | T T | λx.T | T t |↑R T |↓R TAnd we introduce two deduction rules.foldΓ ` pi : ϕΓ `↑Rpi : P andunfoldΓ ` pi : PΓ `↓Rpi : ϕWe also inroduce the following reduction rule.↓R↑Rpi B pi2.3 Supernatural deductionSupernatural deduction is to goes one step further and incorporates in these rulesthe introductions and eliminations for all the connectives of ϕ. More formally,we define the super-rules as follows.Super-rules computationDefinition 1 (Computation of the introduction super-rules). Considera rewrite rule R : P → ϕ. Consider a sequence of variables l = x1, x2, . . . thatdo not occur in the rule. We associate to R an introduction rule of the formpremiseI(Γ, ϕ, l)Γ ` P cond(Γ ,ϕ,l)Where the sequent premiseI(Γ, ϕ, l) and the condition cond(Γ, ϕ, l) are definedby induction on the stucture of ϕ as follows– if ϕ is atomic, then premiseI(Γ, ϕ, l) = (Γ ` ϕ) and cond(Γ, ϕ, l) = ∅,– if ϕ = ϕ1 ⇒ ϕ2 then premiseI(Γ, ϕ, l) = premiseI((Γ, ϕ1), ϕ2, l) andcond(Γ, ϕ, l) = cond((Γ, ϕ1), ϕ2, l)– if ϕ = ∀x ϕ1 then premiseI(Γ, ϕ, y.l) = premiseI(Γ, ϕ1{x := y}, l) andcond(Γ, ϕ, y.l) = cond(Γ, ϕ1{x := y}, l) ∪ {y 6∈ FV(Γ )}Definition 2 (Computation of the elimination super-rules). Considera rewrite rule R : P → ϕ. Consider a sequence of names l = t1, t2, . . .. Weassociate to R an elimination rule of the formΓ ` P premisesE(Γ, ϕ, l)conclusion(Γ, ϕ, l)Where the multiset of sequents premisesE(Γ, ϕ, l) and the sequent conclusion(Γ, ϕ)are defined by induction on the stucture of ϕ as follows– if ϕ is atomic then premisesE(Γ, ϕ, l) = ∅ and conclusion(Γ, ϕ) = (Γ ` ϕ)– if ϕ = ϕ1 ⇒ ϕ2 then premisesE(Γ, ϕ, l) = {Γ ` ϕ1} ∪ premisesE(Γ, ϕ2, l)and conclusion(Γ, ϕ, l) = conclusion(Γ, ϕ2, l)– if ϕ = ∀x ϕ1 then let premisesE(Γ, ϕ, t.l) = premisesE(Γ, ϕ1{x := t}, l)and conclusion(Γ, ϕ, t.l) = conclusion(Γ, ϕ1{x := t}, l)Example 1 (Super-rules for inclusion definition). Given the rewrite rule ⊆: X ⊆Y → ∀z.(z ∈ X ⇒ z ∈ Y ), the associated super deduction rules are:(⊆I) Γ, z ∈ X ` z ∈ YΓ ` X ⊆ Y z /∈ FV(Γ ) (⊆E)Γ ` X ⊆ Y Γ ` t ∈ XΓ ` t ∈ YA proof-term language for Supernatural deduction The proof-term lan-guage is that of proof-terms for predicate logic, enhanced with a pattern languageand the corresponding abstraction:T ::= α | λα : A.T | T T | λx.T | T t | λR(m).T | T R(t)The variables x, y, . . . are variables of the theory while α, β, . . . are proofvariables. The two last constructs allow to interprete the super-rules. In thepattern R(m) the constructor R is applied to a sequence of variables that maybe either term or proof variables and in the term R(t) it is applied to a sequenceof terms that may be either terms of the theory or proof-terms.We can now define the typing rules that correspond to the super-rules above.Definition 3. The arity of a formula ϕ is a sequence of ∀ and⇒ symbols definedby induction on ϕ as follows– if ϕ is atomic arity(ϕ) = [ ],– if ϕ = ϕ1 ⇒ ϕ2 then arity(ϕ) = (⇒ .(arity(ϕ2))),– if ϕ = ∀x ϕ1 then arity(ϕ) = (∀.(arity(ϕ1))).Let ϕ be a formula, a sequence for ϕ is a sequence of distinct variables such thatthe n-th variable of the sequence is a proof variable if the n-th element of thearity of ϕ is ⇒ and a term variable otherwise.Definition 4 (Computation of the abstraction rules). Consider a rewriterule R : P → ϕ. Consider a sequence l for ϕ of variables that do not occur inthe rule. We associate to R an abstraction rule of the formpremiseI(Γ, ϕ, l)Γ ` (λR(l).pi) : P cond(Γ ,ϕ,l)Where the sequent premiseI(Γ, ϕ, l) and the condition cond(Γ, ϕ, l) are definedby induction on the stucture of ϕ as follows– if ϕ is atomic, then premiseI(Γ, ϕ, l) = (Γ ` pi : ϕ) and cond(Γ, ϕ, l) = ∅,– if ϕ = ϕ1 ⇒ ϕ2 then premiseI(Γ, ϕ, α.l) = premiseI((Γ, α : ϕ1), ϕ2, l) andcond(Γ, ϕ, α.l) = cond((Γ, ϕ1), ϕ2, l)– if ϕ = ∀x ϕ1 then premiseI(Γ, ϕ, y.l) = premiseI(Γ, ϕ1{x := y}, l) andcond(Γ, ϕ, y.l) = cond(Γ, ϕ1{x := y}, l) ∪ {y 6∈ FV(Γ )}Definition 5 (Computation of the application rules). Consider a rewriterule R : P → ϕ. Consider a sequence l for ϕ of names. We associate to R theapplication rule of the formΓ ` pi : P premisesE(Γ, ϕ, l)conclusion(Γ, (pi R(l)), ϕ, l)Where the multiset of sequents premisesE(Γ, ϕ, l) and the sequent conclusion(Γ, pi′, ϕ, l)are defined by induction on the stucture of ϕ as follows– if ϕ is atomic then premisesE(Γ, ϕ, l) = ∅ and conclusion(Γ, pi′, ϕ, l) =(Γ ` pi′ : ϕ)– if ϕ = ϕ1 ⇒ ϕ2 then premisesE(Γ, ϕ, τ.l) = {Γ ` τ : ϕ1}∪premisesE(Γ, ϕ2, l)and conclusion(Γ, pi′, ϕ, τ.l) = conclusion(Γ, pi′, ϕ2, l)– if ϕ = ∀x ϕ1 we let premisesE(Γ, ϕ, t.l) = premisesE(Γ, ϕ1{x := t}, l) andconclusion(Γ, pi′, ϕ, t.l) = conclusion(Γ, pi′, ϕ1{x := t}, l)Example 2 (Proof-terms for the inclusion). Our definition of ⊆ uses a witnessand charges an assumption into the context. Thus, the associated proof-termsare those given by the following typing rules:(⊆I) Γ, α:(x ∈ X) ` A : (x ∈ Y )Γ ` λ⊆(x, α).A : (X ⊆ Y ) (⊆E)Γ ` A : (X ⊆ Y ) Γ ` B : (t ∈ X)Γ ` A⊆(t, B) : (t ∈ Y )Definition 6 (Generalized cut elimination). The elimination of a gener-alized cut is represented by a reduction which transmits the witnesses and thelemmas to the proof.λR(m).pi R(t)Bρ pi{m := t}When seeing Supernatural deduction proof-terms as very simple ρ-terms of therewriting calculus, the generalized cut elimination is then a ρ-reduction step.Hence the notation.3 NormalizationWe prove that strong normalization with respect to a rewrite system R is equiv-alent in SND and in fold/unfold. To do so, we introduce translations beetweenfrom each system to the other, such that if pi BS1 pi′, then JpiK B+S2 Jpi′K. Theexistence of such a translation is a sufficient condition for the normalization inS2 to imply that in S1.3.1 From fold/unfold to SNDDefinition 7. To each rule R : P → ϕ, we associate two proof-terms in Super-natural deduction of type ϕ⇒ P and P ⇒ ϕχR↓ = λf λR(m1, . . . ,mn).(f m1 . . .mn)andχR↑ = λp λm1 . . . λmn.(p R(m1, . . . ,mn))where m1, . . . ,mn is a sequence for ϕ.Proposition 1. The term χR↑ (χR↓ x) reduces in two steps to λm1 . . . λmn.(x m1 . . .mn).Definition 8 (Translation). To each proof pi of a formula ϕ in fold/unfold,we associate a proof of ϕ in Supernatural deduction by induction on the structureof pi as follows.– if α is a variable then [α] = α– [λα : A.pi′] = λα : A.[pi′]– [λx.pi′] = λx.[pi′]– [(pi1 pi2)] = [pi1] [pi2]– [(pi′ t)] = [pi′] t– [↑Rpi] = χR↑ [pi]– [↓Rpi] = χR↓ [pi]Definition 9 (another translation). Let pi a proof of a formula ϕ and m1, . . . ,mna sequence for ϕ. We let JpiK = [pi] m1 . . .mn.Proposition 2. If pi is well-typed in Fold, then JpiK is well-typed in Supernaturaldeduction.Proof. By induction on the structure of pi we prove that [pi] is well-typed andhas the same type than pi. Let us detail the case of the fold. The type of pi is Aand the type of ↑Rpi is P . [pi] has type A. χR↑ has type A⇒ P . Then χR↓ [pi] hastype P . Thus JpiK is well-typed.Proposition 3. Let pi and pi′ be two proofs in fold such that piBβ+fold pi′, thenJpiKB+ρ Jpi′K where m1 . . .mn is a sequence for the type of pi.Proof. The only non-trivial case is that of the fold-unfold cut. J↑R↓R piK =χR↑ (χR↓ [pi]) m1 . . .mn Bρ λm1 . . . λmn([pi]m1 . . .mn) m1 . . .mn Bρ JpiK.3.2 From SND to DMDefinition 10 (Translation). To each proof pi of a formula ϕ in SND, weassociate a proof of ϕ in fold/unfold by induction on the structure of pi as follows.– if α is a variable then JαK = α– if t is a term then JtK = t– Jλα : A.pi′K = λα : A.Jpi′K– Jλx.pi′K = λx.Jpi′K– J(pi1 pi2)K = Jpi1K Jpi2K– J(pi′ t)K = Jpi′K t– JλR(x1, . . . , xn).piK =↑R (λx1 . . . xn JpiK)– J(pi R(x1, . . . , xn))K = (↓R JpiK x1 . . . xn)We leave λ-terms as usual and call τ the translation which translates:– an abstraction λR(m).pi as the λ-proof-term corresponding to the same in-troductions in Deduction modulo.– an application pi R(t) as the λ-proof-term corresponding to the same elimi-nations in Deduction modulo.Proposition 4. If pi is well-typed in SND, then JpiK is well-typed in fold/unfold.Proof. By induction on the structure of pi.Proposition 5. If pi Bρ pi′ then JpiKB+β Jpi′K.Proof. The only non-trivial case is that of the SND cut. J(λR(x1, . . . , xn) pi)R(t1, . . . , tn)K =(↓R↑Rλx1 . . . xnJpiK) Jt1K . . . JtnK which reduces to JpiK in n+ 1 reduction steps.References1. Dowek, G., Werner, B.: Proof normalization modulo. Journal of Symbolic Logic68(4) (2003) 1289–13162. Hermant, O.: Me´thodes Se´mantiques en De´duction Modulo. PhD thesis, Universite´Paris 7 - Denis Diderot (2005)3. Dowek, G.: Truth values algebras and proof normalization. In: TYPES. (2006)110–1244. Houtmann, C.: Cohe´rence de la de´duction surnaturelle. Master’s thesis, E´coleNormale Supe´rieure de Cachan (2006)5. Brauner, P., Houtmann, C., Kirchner, C.: Principles of superdeduction. In: LICS.(2007) 41–506. Dowek, G.: About folding-unfolding cuts and cuts modulo. J. Log. Comput. 11(3)(2001) 419–4297. Prawitz, D.: Natural Deduction, a Proof-theoretical Study. (1965)

Normalization in Supernatural deduction and in Deduction modulo

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HAL Id: inria-00141720https://hal.inria.fr/inria-00141720v3Preprint submitted on 16 Nov 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Normalization in Supernatural deduction and inDeduction moduloPaul Brauner, Gilles Dowek, Benjamin WackTo cite this version:Paul Brauner, Gilles Dowek, Benjamin Wack. Normalization in Supernatural deduction and in De-duction modulo. 2007. ￿inria-00141720v3￿Normalization in Supernatural deduction and inDeduction moduloPaul Brauner1, Gilles Dowek2, Benjamin Wack31 École polytechnique and INRIA,LIX, École polytechnique, 91128 Palaiseau Cedex, Francegilles.dowek@polytechnique.edu http://www.lix.polytechnique.fr/~dowek/2 Loria, Campus Scientifique - BP 239 - 54506 Vandoeuvre-lès-Nancy Cedex, Francepaul.brauner@loria.frhttp://www.loria.fr/~brauner/3 Loria, Campus Scientifique - BP 239 - 54506 Vandoeuvre-lès-Nancy Cedex, FranceAbstract. Deduction modulo and Supernatural deduction are two ex-tentions of predicate logic with computation rules. Whereas the applica-tion of computation rules in deduction modulo is transparent, these rulesare used to build non-logical deduction rules in Supernatural deduction.In both cases, adding computation rules may jeopardize proof normal-ization, but various conditions have been given in both cases, so thatnormalization is preserved. We prove in this paper that normalizationin Supernatural deduction and in Deduction modulo are equivalent, i.e.the set of computation rules for which one system strongly normalizes isthe same as the set of computation rules for which the other is.1 IntroductionIntegrating a theory into a deduction system has several advantages over workingwith axioms. It allows proof-search algorithms to avoid some redundancies, itpermits in some cases to get witnesses from constructive proofs and hence toprogram with proofs, etc.One way to do so is to formulate the theory of interest as a rewrite system andidentify congruent formulae. This is the idea of deduction modulo. For instance,the axiom ∀X∀Y ((X ⊆ Y ) ⇔ ∀z (z ∈ X ⇒ z ∈ Y )) may be formulated as therewrite rule (X ⊆ Y )→ ∀z (z ∈ X ⇒ z ∈ Y ) and(∀I)Γ ` x ∈ A⇒ x ∈ BΓ ` A ⊆ Bis then a valid deduction step. Thus the natural deduction rules have to bereformulated in order to take the congruence into account.An more deterministic alternative to deduction modulo is Supernatural De-duction where the rewrite rules are translated into new deduction rules in thelogic. For instance, the rewrite rule (X ⊆ Y )→ ∀z (z ∈ X ⇒ z ∈ Y ) yields thefollowing introduction and elimination rules∀-introΓ ` (z ∈ X ⇒ z ∈ Y )Γ ` X ⊆ Yz /∈ FV(Γ ) ∀-elimΓ ` X ⊆ YΓ ` t ∈ X ⇒ t ∈ Ywhere all connectors and quantifiers have disappeared. It may be argued thatthese rules are much closer to usual mathematical practice than either predicatelogic with axioms or Deduction Modulo.Since soundness and completeness with respect to predicate logic with axiomshas been proved for both systems, they prove the same theorems but we areinterested in a stronger equivalence property.To explain this strong equivalence property, we must first notice that in bothcases, proof normalization and cut elimination are jeopardized. However, variousconditions have been given so that normalization is preserved. For deductionmodulo, such criteria have been developed in [1,2,3]. For SND, criteria havebeen developed in [4].This raises the following problem: are there some rewrite system for whichone deduction system has the proof normalization property but not the other ordoes proof normalization depend only on the rewrite system itself ? We provein this paper that this is indeed the case: given an orthogonal rewrite system Rrewriting atomic formulae to arbitrary formulae, proofs normalize in DeductionModulo(R) if and only if they normalize in Supernatural Deduction(R). As aconsequence, the semantic criteria developed for Deduction Modulo [3] may betransferred to supernatural deduction.Our result is limited to minimal logic (i.e. our connectives are restricted toimplication and universal quantification). The result can be extended to con-jonction but the extension to the disjunction, the existential quantifier and con-tradiction is more challenging. Indeed, Supernatural Deduction rules are alreadytrickier to define when such connectors and quantifiers are involved. A possiblesolution may be to move from natural deduction to sequent calculus [5].To prove this equivalence between Supernatural Deduction and DeductionModulo, we shall prove that they are both equivalent to a third and older systemdue to Prawitz: Natural Deduction with folding and unfolding rules. Actually, theequivalence of Deduction modulo and Prawitz’ system has already been provedin [6]. Thus, all that remains to be done is to prove the equivalence betweenPrawitz’ system and Supernatural Deduction. Fortunately, this can be achievedby a simple syntactic translation of proofs.This paper contributes to show that the choice of a formalism or anotherseems to be a rather superficial choice, as properties such as strong normalizationare quite stable when one switches from one system to another.2 From fold/unfold to supernatural deduction2.1 Natural deductionOur starting point is natural deduction for minimal logic with five rules: (Ax),(⇒I), (⇒E), (∀I), (∀E). We use a lambda calculus notation for proofs:T ::= α | λα : A.T | T T | λx.T | T tThe deduction rules enriched with proof-terms are the following: but these rulesare typed with different typing rules(Ax)Γ, α : A ` α : A(⇒I)Γ, α : A ` π : BΓ ` (λα : A.π) : A⇒ B(⇒E)Γ ` π1 : A⇒ B Γ ` π2 : AΓ ` (π1 π2) : B(∀I)Γ ` π : AΓ ` (λx.π) : ∀xAx /∈ FV(Γ ) (∀E)Γ ` π : ∀xAΓ ` (π t) : A{x := t}The reduction rules on these proof-terms are the two kinds of β-reduction((λα : A π1) π2) B π1{α := π2}((λx π) t) B π{x := t}2.2 Natural deduction with folding and unfolding rulesConsider an orthogonal rewrite system R rewriting atomic formulae to formulae.For a rewrite rule P → ϕ, we may add Prawitz’ folding and unfolding rules[7] replacing ϕ by P and P by ϕ. To do so, we enrich the proof-terms language.T ::= α | λα : A.T | T T | λx.T | T t |↑R T |↓R TAnd we introduce two deduction rules.foldΓ ` π : ϕΓ `↑Rπ : P andunfoldΓ ` π : PΓ `↓Rπ : ϕWe also inroduce the following reduction rule.↓R↑Rπ B π2.3 Supernatural deductionSupernatural deduction is to goes one step further and incorporates in these rulesthe introductions and eliminations for all the connectives of ϕ. More formally,we define the super-rules as follows.Super-rules computationDefinition 1 (Computation of the introduction super-rules). Considera rewrite rule R : P → ϕ. Consider a sequence of variables l = x1, x2, . . . thatdo not occur in the rule. We associate to R an introduction rule of the formpremiseI(Γ, ϕ, l)Γ ` Pcond(Γ ,ϕ,l)Where the sequent premiseI(Γ, ϕ, l) and the condition cond(Γ, ϕ, l) are definedby induction on the stucture of ϕ as follows– if ϕ is atomic, then premiseI(Γ, ϕ, l) = (Γ ` ϕ) and cond(Γ, ϕ, l) = ∅,– if ϕ = ϕ1 ⇒ ϕ2 then premiseI(Γ, ϕ, l) = premiseI((Γ, ϕ1), ϕ2, l) andcond(Γ, ϕ, l) = cond((Γ, ϕ1), ϕ2, l)– if ϕ = ∀x ϕ1 then premiseI(Γ, ϕ, y.l) = premiseI(Γ, ϕ1{x := y}, l) andcond(Γ, ϕ, y.l) = cond(Γ, ϕ1{x := y}, l) ∪ {y 6∈ FV(Γ )}Definition 2 (Computation of the elimination super-rules). Considera rewrite rule R : P → ϕ. Consider a sequence of names l = t1, t2, . . .. Weassociate to R an elimination rule of the formΓ ` P premisesE(Γ, ϕ, l)conclusion(Γ, ϕ, l)Where the multiset of sequents premisesE(Γ, ϕ, l) and the sequent conclusion(Γ, ϕ)are defined by induction on the stucture of ϕ as follows– if ϕ is atomic then premisesE(Γ, ϕ, l) = ∅ and conclusion(Γ, ϕ) = (Γ ` ϕ)– if ϕ = ϕ1 ⇒ ϕ2 then premisesE(Γ, ϕ, l) = {Γ ` ϕ1} ∪ premisesE(Γ, ϕ2, l)and conclusion(Γ, ϕ, l) = conclusion(Γ, ϕ2, l)– if ϕ = ∀x ϕ1 then let premisesE(Γ, ϕ, t.l) = premisesE(Γ, ϕ1{x := t}, l)and conclusion(Γ, ϕ, t.l) = conclusion(Γ, ϕ1{x := t}, l)Example 1 (Super-rules for inclusion definition). Given the rewrite rule ⊆: X ⊆Y → ∀z.(z ∈ X ⇒ z ∈ Y ), the associated super deduction rules are:(⊆I)Γ, z ∈ X ` z ∈ YΓ ` X ⊆ Yz /∈ FV(Γ ) (⊆E)Γ ` X ⊆ Y Γ ` t ∈ XΓ ` t ∈ YA proof-term language for Supernatural deduction The proof-term lan-guage is that of proof-terms for predicate logic, enhanced with a pattern languageand the corresponding abstraction:T ::= α | λα : A.T | T T | λx.T | T t | λR(m).T | T R(t)The variables x, y, . . . are variables of the theory while α, β, . . . are proofvariables. The two last constructs allow to interprete the super-rules. In thepattern R(m) the constructor R is applied to a sequence of variables that maybe either term or proof variables and in the term R(t) it is applied to a sequenceof terms that may be either terms of the theory or proof-terms.We can now define the typing rules that correspond to the super-rules above.Definition 3. The arity of a formula ϕ is a sequence of ∀ and⇒ symbols definedby induction on ϕ as follows– if ϕ is atomic arity(ϕ) = [ ],– if ϕ = ϕ1 ⇒ ϕ2 then arity(ϕ) = (⇒ .(arity(ϕ2))),– if ϕ = ∀x ϕ1 then arity(ϕ) = (∀.(arity(ϕ1))).Let ϕ be a formula, a sequence for ϕ is a sequence of distinct variables such thatthe n-th variable of the sequence is a proof variable if the n-th element of thearity of ϕ is ⇒ and a term variable otherwise.Definition 4 (Computation of the abstraction rules). Consider a rewriterule R : P → ϕ. Consider a sequence l for ϕ of variables that do not occur inthe rule. We associate to R an abstraction rule of the formpremiseI(Γ, ϕ, l)Γ ` (λR(l).π) : Pcond(Γ ,ϕ,l)Where the sequent premiseI(Γ, ϕ, l) and the condition cond(Γ, ϕ, l) are definedby induction on the stucture of ϕ as follows– if ϕ is atomic, then premiseI(Γ, ϕ, l) = (Γ ` π : ϕ) and cond(Γ, ϕ, l) = ∅,– if ϕ = ϕ1 ⇒ ϕ2 then premiseI(Γ, ϕ, α.l) = premiseI((Γ, α : ϕ1), ϕ2, l) andcond(Γ, ϕ, α.l) = cond((Γ, ϕ1), ϕ2, l)– if ϕ = ∀x ϕ1 then premiseI(Γ, ϕ, y.l) = premiseI(Γ, ϕ1{x := y}, l) andcond(Γ, ϕ, y.l) = cond(Γ, ϕ1{x := y}, l) ∪ {y 6∈ FV(Γ )}Definition 5 (Computation of the application rules). Consider a rewriterule R : P → ϕ. Consider a sequence l for ϕ of names. We associate to R theapplication rule of the formΓ ` π : P premisesE(Γ, ϕ, l)conclusion(Γ, (π R(l)), ϕ, l)Where the multiset of sequents premisesE(Γ, ϕ, l) and the sequent conclusion(Γ, π′, ϕ, l)are defined by induction on the stucture of ϕ as follows– if ϕ is atomic then premisesE(Γ, ϕ, l) = ∅ and conclusion(Γ, π′, ϕ, l) =(Γ ` π′ : ϕ)– if ϕ = ϕ1 ⇒ ϕ2 then premisesE(Γ, ϕ, τ.l) = {Γ ` τ : ϕ1}∪premisesE(Γ, ϕ2, l)and conclusion(Γ, π′, ϕ, τ.l) = conclusion(Γ, π′, ϕ2, l)– if ϕ = ∀x ϕ1 we let premisesE(Γ, ϕ, t.l) = premisesE(Γ, ϕ1{x := t}, l) andconclusion(Γ, π′, ϕ, t.l) = conclusion(Γ, π′, ϕ1{x := t}, l)Example 2 (Proof-terms for the inclusion). Our definition of ⊆ uses a witnessand charges an assumption into the context. Thus, the associated proof-termsare those given by the following typing rules:(⊆I)Γ, α:(x ∈ X) ` A : (x ∈ Y )Γ ` λ⊆(x, α).A : (X ⊆ Y )(⊆E)Γ ` A : (X ⊆ Y ) Γ ` B : (t ∈ X)Γ ` A⊆(t, B) : (t ∈ Y )Definition 6 (Generalized cut elimination). The elimination of a gener-alized cut is represented by a reduction which transmits the witnesses and thelemmas to the proof.λR(m).π R(t) Bρ π{m := t}When seeing Supernatural deduction proof-terms as very simple ρ-terms of therewriting calculus, the generalized cut elimination is then a ρ-reduction step.Hence the notation.3 NormalizationWe prove that strong normalization with respect to a rewrite system R is equiv-alent in SND and in fold/unfold. To do so, we introduce translations beetweenfrom each system to the other, such that if π BS1 π′, then JπK B+S2 Jπ′K. Theexistence of such a translation is a sufficient condition for the normalization inS2 to imply that in S1.3.1 From fold/unfold to SNDDefinition 7. To each rule R : P → ϕ, we associate two proof-terms in Super-natural deduction of type ϕ⇒ P and P ⇒ ϕχR↓ = λf λR(m1, . . . ,mn).(f m1 . . .mn)andχR↑ = λp λm1 . . . λmn.(p R(m1, . . . ,mn))where m1, . . . ,mn is a sequence for ϕ.Proposition 1. The term χR↑ (χR↓ x) reduces in two steps to λm1 . . . λmn.(x m1 . . .mn).Definition 8 (Translation). To each proof π of a formula ϕ in fold/unfold,we associate a proof of ϕ in Supernatural deduction by induction on the structureof π as follows.– if α is a variable then [α] = α– [λα : A.π′] = λα : A.[π′]– [λx.π′] = λx.[π′]– [(π1 π2)] = [π1] [π2]– [(π′ t)] = [π′] t– [↑Rπ] = χR↑ [π]– [↓Rπ] = χR↓ [π]Definition 9 (another translation). Let π a proof of a formula ϕ and m1, . . . ,mna sequence for ϕ. We let JπK = [π] m1 . . .mn.Proposition 2. If π is well-typed in Fold, then JπK is well-typed in Supernaturaldeduction.Proof. By induction on the structure of π we prove that [π] is well-typed andhas the same type than π. Let us detail the case of the fold. The type of π is Aand the type of ↑Rπ is P . [π] has type A. χR↑ has type A⇒ P . Then χR↓ [π] hastype P . Thus JπK is well-typed.Proposition 3. Let π and π′ be two proofs in fold such that πBβ+fold π′, thenJπK B+ρ Jπ′K where m1 . . .mn is a sequence for the type of π.Proof. The only non-trivial case is that of the fold-unfold cut. J↑R↓R πK =χR↑ (χR↓ [π]) m1 . . .mn Bρ λm1 . . . λmn([π]m1 . . .mn) m1 . . .mn Bρ JπK.3.2 From SND to DMDefinition 10 (Translation). To each proof π of a formula ϕ in SND, weassociate a proof of ϕ in fold/unfold by induction on the structure of π as follows.– if α is a variable then JαK = α– if t is a term then JtK = t– Jλα : A.π′K = λα : A.Jπ′K– Jλx.π′K = λx.Jπ′K– J(π1 π2)K = Jπ1K Jπ2K– J(π′ t)K = Jπ′K t– JλR(x1, . . . , xn).πK =↑R (λx1 . . . xn JπK)– J(π R(x1, . . . , xn))K = (↓R JπK x1 . . . xn)We leave λ-terms as usual and call τ the translation which translates:– an abstraction λR(m).π as the λ-proof-term corresponding to the same in-troductions in Deduction modulo.– an application π R(t) as the λ-proof-term corresponding to the same elimi-nations in Deduction modulo.Proposition 4. If π is well-typed in SND, then JπK is well-typed in fold/unfold.Proof. By induction on the structure of π.Proposition 5. If π Bρ π′ then JπK B+β Jπ′K.Proof. The only non-trivial case is that of the SND cut. J(λR(x1, . . . , xn) π)R(t1, . . . , tn)K =(↓R↑Rλx1 . . . xnJπK) Jt1K . . . JtnK which reduces to JπK in n+ 1 reduction steps.References1. Dowek, G., Werner, B.: Proof normalization modulo. Journal of Symbolic Logic68(4) (2003) 1289–13162. Hermant, O.: Méthodes Sémantiques en Déduction Modulo. PhD thesis, UniversitéParis 7 - Denis Diderot (2005)3. Dowek, G.: Truth values algebras and proof normalization. In: TYPES. (2006)110–1244. Houtmann, C.: Cohérence de la déduction surnaturelle. Master’s thesis, ÉcoleNormale Supérieure de Cachan (2006)5. Brauner, P., Houtmann, C., Kirchner, C.: Principles of superdeduction. In: LICS.(2007) 41–506. Dowek, G.: About folding-unfolding cuts and cuts modulo. J. Log. Comput. 11(3)(2001) 419–4297. Prawitz, D.: Natural Deduction, a Proof-theoretical Study. (1965)

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Normalization in Supernatural deduction and in Deduction modulo

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