## Complete sequent calculi for induction and infinite descent (2007)

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Venue: | Proceedings of LICS-22 |

Citations: | 18 - 6 self |

### BibTeX

@INPROCEEDINGS{Brotherston07completesequent,

author = {James Brotherston},

title = {Complete sequent calculi for induction and infinite descent},

booktitle = {Proceedings of LICS-22},

year = {2007},

pages = {51--60}

}

### OpenURL

### Abstract

This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing inductively defined predicates on the left of sequents. We show this system to be cut-free complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system uses infinite (non-well-founded) proofs to represent arguments by infinite descent. In this system, the left rules for inductively defined predicates are simple case-split rules, and an infinitary, global condition on proof trees is required to ensure soundness. We show this system to be cut-free complete with respect to standard models, and again infer the eliminability of cut. The second infinitary system is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic ” system subsumes the first system for proof by induction. We conjecture that the two systems are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.

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Citation Context ...ant on the left and on the right of sequents respectively. Gentzen’s well-known cut-elimination theorem implies that direct proofs, using these rules alone, are sufficient to derive any valid sequent =-=[10]-=-. In addition to its theoretical elegance, this has implications for proof search, with the locally applicable proof rules thereby constrained by the logical constants appearing in the current goal. H... |

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Citation Context ...finite paths in GP such that there does not exist an infinitely progressing trace on any tail of the path. P is then a CLKID ω proof exactly if B accepts no strings, which is a decidable problem (cf. =-=[39]-=-). The full construction appears in appendix A of [4]. Similar arguments also appear in [27, 34, 21]. ✷ We now turn our attention to the question of the relationship between CLKID ω and our system for... |

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Citation Context ... 4.9 we are aware of in the literature appears in [15]. There, certain refutations are defined, which can be seen as providing an analogous proof system to LKID ω for Kozen’s propositional µ-calculus =-=[11]-=-. Indeed, refutations are formulated using a trace-based proof condition very similar to Defn. 4.5. (Other similar conditions appear in [12, 19].) One of the main results of [15] is a completeness the... |

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Citation Context ...urse, the use of such semantic methods to establish cut-eliminability is not new. For example, the original proof of Takeuti’s Conjecture (the eliminability of cut in second-order logic) was semantic =-=[28, 11]-=-.SEQUENT CALCULI FOR INDUCTION AND INFINITE DESCENT 3 However, compared with the semantic proof of Takeuti’s Conjecture, the class of Henkin models we consider seems a reasonably natural class of str... |

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Citation Context ...ction in Φ, will always use the explicit format of (Def) above. 1 The inclusion of equality in the language is a minor departure from [2]. The standard interpretation of the inductive predicates (cf. =-=[1]-=-) is obtained as usual by considering prefixed points of a monotone operator constructed from the definition set Φ. For standard models, this least fixed point can be constructed in iterative approxim... |

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Citation Context ...proof system to LKID ω for Kozen’s propositional µ-calculus [11]. Indeed, refutations are formulated using a trace-based proof condition very similar to Defn. 4.5. (Other similar conditions appear in =-=[12, 19]-=-.) One of the main results of [15] is a completeness theorem for refutations. Nevertheless, the situations are quite different in many respects. The proposition µ-calculus is decidable, whereas validi... |

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Citation Context ...l component of proof assistants and theorem provers. Often, libraries are provided containing collections of useful induction principles associated with a given set of inductive definitions, see e.g. =-=[16, 9, 18]-=-. In other cases, mechanisms permitting “cyclic” proof arguments are used, with intricate conditions imposed to ensure soundness, see e.g. [23, 17, 8]. ∗ Research undertaken while a PhD student at LFC... |

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Citation Context ...th a given set of inductive definitions, see e.g. [16, 9, 18]. In other cases, mechanisms permitting “cyclic” proof arguments are used, with intricate conditions imposed to ensure soundness, see e.g. =-=[23, 17, 8]-=-. ∗ Research undertaken while a PhD student at LFCS, School of Informatics, University of Edinburgh, supported by an EPSRC PhD studentship. † Research supported by an EPSRC Advanced Research Fellowshi... |

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Citation Context ...kin completeness of LKID). If Γ ⊢ ∆ is Henkin valid, then it is cut-free provable in LKID. Proof. The proof is an extension of the direct style of completeness proof for Gentzen’s LK as given in e.g. =-=[8]-=-. Briefly, supposing that Γ ⊢ ∆ is not cutfree provable in LKID, one uses a uniform proof-search procedure to construct a sequence of underivable sequents Γi ⊢ ∆i, which can together be used to build ... |

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Citation Context ...ch are paths in GP such that there does not exist an infinitely progressing trace on any tail of the path. P is then a CLKω ID proof exactly if B accepts no strings, which is a decidable problem (cf. =-=[29]-=-). The full construction appears in appendix A of [3]. Similar arguments also appear in [21, 25, 16]. ✷ We now turn our attention to the question of the relationship between CLKω ID and our system for... |

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Citation Context ...es in inductive arguments (an issue which causes serious trouble for theorem provers [7]). Nevertheless, cut-eliminability for LKID is potentially a useful property for constraining proof search; see =-=[19]-=- for related discussion in the intuitionistic case. Also, one can show that the eliminability of cut in LKID implies the consistency of Peano arithmetic, so there can be no straightforward combinatori... |

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51 | D.: Rudiments of µ-Calculus
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Citation Context ...erating any complete subclass of recursive proofs. Hence LKID ω is (unsurprisingly) not suitable for formal reasoning. The closest analogue of Theorem 4.9 we are aware of in the literature appears in =-=[15]-=-. There, certain refutations are defined, which can be seen as providing an analogous proof system to LKID ω for Kozen’s propositional µ-calculus [11]. Indeed, refutations are formulated using a trace... |

48 |
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(Show Context)
Citation Context ...urse, the use of such semantic methods to establish cut-eliminability is not new. For example, the original proof of Takeuti’s Conjecture (the eliminability of cut in second-order logic) was semantic =-=[28, 11]-=-.SEQUENT CALCULI FOR INDUCTION AND INFINITE DESCENT 3 However, compared with the semantic proof of Takeuti’s Conjecture, the class of Henkin models we consider seems a reasonably natural class of str... |

42 | The automation of proof by mathematical induction
- Bundy
(Show Context)
Citation Context ...ules. This is an unavoidable phenomenon, and corresponds to the well-known need for generalising induction hypotheses in inductive arguments (an issue which causes serious trouble for theorem provers =-=[8]-=-). Nevertheless, cut-eliminability for LKID is potentially a useful property for constraining proof search; see [24] for related discussion in the intuitionistic case. There are two natural questions ... |

33 | A logical framework for reasoning about logical specifications
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Citation Context ...as left-introduction rules in sequent calculus. Nonetheless, it is only recently that sequent calculus counterparts of Martin-Löf’s system have been explicitly considered, by McDowell, Miller and Tiu =-=[14, 21]-=-. Ours is a simple classical analogue of these intuitionistic systems. Our main new result about LKID is a completeness result, for cut-free proofs, relative to a natural class of “Henkin models” for ... |

31 |
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Citation Context ...on results are impossible in the presence of inductive definitions. In fact, the real limitation is not the impossibility of cut-elimination, but rather that the subformula property is not achievable =-=[20]-=-, and indeed the subformula property does not hold for cut-free proofs in LKID. In fact, the possibility of obtaining a full cut-elimination result is not surprising if one is familiar with the aforem... |

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Citation Context ...ulus refutations [15] and proofs in CLKID ω . On the other hand, there is an analogy between proofs in Kozen’s system, and proofs in LKID. Walukiewicz’ solution to the µ-calculus completeness problem =-=[22]-=- established the equivalence of the two, but it is far from clear whether similar methods are applicable in our setting. 7. Future work One direction for further research is to investigate whether mor... |

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Citation Context ...les in sequent calculus. Nonetheless, it is only relatively recently that sequent calculus counterparts of Martin-Löf’s system have been explicitly considered, by McDowell, Miller, Momigliano and Tiu =-=[19, 20, 30]-=-. Ours is a natural classical analogue of these intuitionistic systems. For LKID, we prove soundness and completeness (the latter for the cut-free subsystem), relative to a natural class of “Henkin mo... |

22 |
Inductive definitions: automation and application
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(Show Context)
Citation Context ...l component of proof assistants and theorem provers. Often, libraries are provided containing collections of useful induction principles associated with a given set of inductive definitions, see e.g. =-=[16, 9, 18]-=-. In other cases, mechanisms permitting “cyclic” proof arguments are used, with intricate conditions imposed to ensure soundness, see e.g. [23, 17, 8]. ∗ Research undertaken while a PhD student at LFC... |

20 | Equality reasoning in sequentbased calculi
- Degtyarev, Voronkov
- 2001
(Show Context)
Citation Context ...s is applied to all formulas in Γ. We use the standard sequent calculus rules as given in many sources (see e.g. [7, 5]), together with the following rules for explicit substitution and equality, cf. =-=[6]-=-. Γ ⊢ ∆ (Subst) Γ[θ] ⊢ ∆[θ] (=R) Γ ⊢ t = t, ∆ Γ[u/x, t/y] ⊢ ∆[u/x, t/y] (=L) Γ[t/x, u/y],t= u ⊢ ∆[t/x, u/y] To these rules we add rules for introducing inductive predicates on the left and right of se... |

20 |
Derivation and use of induction schemes in higher-order logic
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- 1997
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Citation Context ...l component of proof assistants and theorem provers. Often, libraries are provided containing collections of useful induction principles associated with a given set of inductive definitions, see e.g. =-=[16, 9, 18]-=-. In other cases, mechanisms permitting “cyclic” proof arguments are used, with intricate conditions imposed to ensure soundness, see e.g. [23, 17, 8]. ∗ Research undertaken while a PhD student at LFC... |

18 | On the structure of inductive reasoning: Circular and tree-shaped proofs in the mu-calculus
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- 2003
(Show Context)
Citation Context ...it as a conjecture. Conjecture 6.5 ([2]). If there is a CLKID ω proof of Γ ⊢ ∆ then there is an LKID proof of Γ ⊢ ∆. This conjecture does not seem straightforward. For example, the methods applied in =-=[20]-=-, which show, in a different setting, the equivalence of a weaker global proof condition with a local transfinite induction principle, do not adapt. The difficulties are reminiscent of those in provin... |

17 |
Enhancing program verification with lemmas
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(Show Context)
Citation Context ...een adapted theoretically to the setting of the bunched logic BI [5] and to Hoare-style termination proofs in separation logic [6], and is also beginning to see practical use in theorem proving tools =-=[26, 41]-=-. Plausibly, cyclic reasoning is also likely to prove especially useful for demonstrating properties of mutually defined relations, for which the associated induction principles are often extremely co... |

16 | A calculus of circular proofs and its categorical semantics
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(Show Context)
Citation Context ...ich model a propositional logic of linear conjunction and disjunction with fixed points), and of related categories with fixed points (called Ω-models), both of which have been studied by Santocanale =-=[30, 29]-=-. Two main properties distinguish the aforementioned proof systems for propositional fixed point logics, influenced by refutations, from the first-order systems of the present paper. First, for all th... |

14 | Cyclic proofs for first-order logic with inductive definitions
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- 2005
(Show Context)
Citation Context ...ce, this has implications for proof search, with the locally applicable proof rules thereby constrained by the logical constants appearing in the current goal. In a previous paper by the first author =-=[2]-=-, Gentzen’s LK was extended to obtain two proof systems for classical first-order logic with inductively defined predicates: (i) a system containing explicit proof rules embodying the standard inducti... |

13 |
Descente Infinie + Deduction
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- 2004
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Citation Context ...th a given set of inductive definitions, see e.g. [16, 9, 18]. In other cases, mechanisms permitting “cyclic” proof arguments are used, with intricate conditions imposed to ensure soundness, see e.g. =-=[23, 17, 8]-=-. ∗ Research undertaken while a PhD student at LFCS, School of Informatics, University of Edinburgh, supported by an EPSRC PhD studentship. † Research supported by an EPSRC Advanced Research Fellowshi... |

12 |
An introduction to inductive definitions. In: Handbook of mathematical logic
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Citation Context ...ive definition set Φ for Σ and, when we need to consider an arbitrary production in Φ, will always use the explicit format of (Def) above. The standard interpretation of the inductive predicates (cf. =-=[1]-=-) is obtained as usual by considering prefixed points of a monotone operator constructed from the definition set Φ. For standard models, the least prefixed point of this operator can be constructed in... |

12 | C.: Cyclic proofs of program termination in separation logic
- Brotherston, Bornat, et al.
(Show Context)
Citation Context ...setting of O’Hearn and Pym’s BI [4], and subsequently mooted as a technique for provingSEQUENT CALCULI FOR INDUCTION AND INFINITE DESCENT 25 termination of imperative programs using separation logic =-=[5]-=-. Plausibly, cyclic reasoning is also likely to prove especially useful for demonstrating properties of mutually defined relations, for which the associated induction principles are often extremely co... |

11 | A proof system for the linear time µ-calculus
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Citation Context ... system to LKID ω for Kozen’s propositional µ-calculus [19]. Indeed, refutations are formulated using a trace-based proof condition very similar to Definition 5.5. (Other similar conditions appear in =-=[10, 21, 34, 36, 42]-=-.) One of the main results of [27] is a completeness theorem for refutations. Nevertheless, the situations are quite different. In particular, the propositional µ-calculus is decidable, whereas (stand... |

10 | Free µ-lattices
- Santocanale
- 2002
(Show Context)
Citation Context ...ich model a propositional logic of linear conjunction and disjunction with fixed points), and of related categories with fixed points (called Ω-models), both of which have been studied by Santocanale =-=[30, 29]-=-. Two main properties distinguish the aforementioned proof systems for propositional fixed point logics, influenced by refutations, from the first-order systems of the present paper. First, for all th... |

8 |
Proof Theory and Logical Complexity, Volume 1. bibliopolis
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- 1987
(Show Context)
Citation Context ...s, and use the notation Γ[θ] to mean that the substitution θ of terms for free variables is applied to all formulas in Γ. We use the standard sequent calculus rules as given in many sources (see e.g. =-=[7, 5]-=-), together with the following rules for explicit substitution and equality, cf. [6]. Γ ⊢ ∆ (Subst) Γ[θ] ⊢ ∆[θ] (=R) Γ ⊢ t = t, ∆ Γ[u/x, t/y] ⊢ ∆[u/x, t/y] (=L) Γ[t/x, u/y],t= u ⊢ ∆[t/x, u/y] To these... |

8 | On global induction mechanisms in a µ-calculus with explicit approximations
- Sprenger, Dam
(Show Context)
Citation Context ...proof system to LKID ω for Kozen’s propositional µ-calculus [11]. Indeed, refutations are formulated using a trace-based proof condition very similar to Defn. 4.5. (Other similar conditions appear in =-=[12, 19]-=-.) One of the main results of [15] is a completeness theorem for refutations. Nevertheless, the situations are quite different in many respects. The proposition µ-calculus is decidable, whereas validi... |

8 | Completions of µ-algebras
- Santocanale
(Show Context)
Citation Context ...h these coincidences can be viewed as consequences of the fact that free algebraic structures with fixedpoints can be embedded into corresponding complete lattices, a perspective that is developed in =-=[30, 31]-=-. 31free transfinite induction up to ǫ0 (the use of transfinite induction is necessary because cutelimination for LKID implies the consistency of PA, as demonstrated by our Theorem 3.15). In the case... |

7 | On the proof theory of the modal mu-calculus
- Studer
(Show Context)
Citation Context ... system to LKID ω for Kozen’s propositional µ-calculus [19]. Indeed, refutations are formulated using a trace-based proof condition very similar to Definition 5.5. (Other similar conditions appear in =-=[10, 21, 34, 36, 42]-=-.) One of the main results of [27] is a completeness theorem for refutations. Nevertheless, the situations are quite different. In particular, the propositional µ-calculus is decidable, whereas (stand... |

6 |
The Mathematical Career of Pierre de Fermat, 1601–1665
- Mahoney
- 1994
(Show Context)
Citation Context ... imposed to ensure soundness, see e.g. [32, 23, 12]. These conditions can be broadly construed as versions of the well-known mathematical principle of infinite descent originally formalised by Fermat =-=[17]-=-. In this article we develop proof-theoretic foundations for this infinite descent style of inductive reasoning, and compare them with the corresponding (but quite different) foundations for proof by ... |

5 |
Sequent Calculus Proof Systems for Inductive Definitions
- Brotherston
- 2006
(Show Context)
Citation Context ...e apparent difficulties its proof poses. Due to space constraints, only outline proofs of our main results are included in this paper. Full proofs can be found in the first author’s recent PhD thesis =-=[3]-=-. 2. Syntax and semantics of first-order logic with inductive definitions (FOL ID) In this section we give the syntax and semantics of classical first-order logic with inductively defined predicates,s... |

4 | Formalised inductive reasoning in the logic of bunched implications
- Brotherston
- 2007
(Show Context)
Citation Context ..., as formulated in the present paper, does have potential applications. The style of cyclic reasoning we have developed for FOLID has been adapted theoretically to the setting of the bunched logic BI =-=[5]-=- and to Hoare-style termination proofs in separation logic [6], and is also beginning to see practical use in theorem proving tools [26, 41]. Plausibly, cyclic reasoning is also likely to prove especi... |

4 | Duality for modal µ-logics
- Hartonas
- 1998
(Show Context)
Citation Context ...pke-frame models of the modal µ-calculus, also considered in op. cit. Our more general notion of Henkin model also has an analogue, for the modal µ-calculus, given by the modal µ-frames introduced in =-=[2, 16]-=-. In these last references, the completeness of Kozen’s axiomatization is established relative to modal µ-frames, and our Theorem 3.6 can be seen as analogous to these results. 2 Independently to our ... |

4 |
Derivation and use of induction schemes in higher-orderlogic
- Slind
- 1997
(Show Context)
Citation Context ...al component of proof assistants and theorem provers. Often, libraries are provided containing collections of useful induction principles associated with a given set of inductive definitions, see e.g.=-=[28, 15, 33]-=-. In other cases, mechanisms permitting “cyclic” proof arguments are used, with intricate conditions imposed to ensure soundness, see e.g. [44, 32, 14]. These conditions can be broadly construed as ve... |

3 |
Poítin: Distilling theorems from conjectures
- Hamilton
- 2005
(Show Context)
Citation Context ...th a given set of inductive definitions, see e.g. [16, 9, 18]. In other cases, mechanisms permitting “cyclic” proof arguments are used, with intricate conditions imposed to ensure soundness, see e.g. =-=[23, 17, 8]-=-. ∗ Research undertaken while a PhD student at LFCS, School of Informatics, University of Edinburgh, supported by an EPSRC PhD studentship. † Research supported by an EPSRC Advanced Research Fellowshi... |

3 |
Voronkov: Equality Reasoning in Sequent-Based Calculi. Handbook of Automated Reasoning vol
- Degtyarev, A
- 2001
(Show Context)
Citation Context ... of a rule instance, we mean the distinguished formula that is introduced by the rule into its conclusion. Somewhat unusually, we include a rule for explicit substitution, and rules for equality (cf. =-=[9]-=-). Although these rules are inessential inclusions in LKID, they will prove useful in our infinitary proof systems for infinite descent in FOLID, introduced later. To the rules in Figure 1 we add rule... |

2 |
The Automation of Proof by Mathematical Induction”, Handbook of Automated Reasoning 2001: 845-911 [Buss
- Bundy
- 1998
(Show Context)
Citation Context ...ules. This is an unavoidable phenomenon, and corresponds to the well-known need for generalising induction hypotheses in inductive arguments (an issue which causes serious trouble for theorem provers =-=[7]-=-). Nevertheless, cut-eliminability for LKID is potentially a useful property for constraining proof search; see [19] for related discussion in the intuitionistic case. Also, one can show that the elim... |

1 |
calculus proof systems for inductive definitions
- Sequent
- 2006
(Show Context)
Citation Context ... our completeness result is possibly of some interest in its own right. Since our results on LKID use standard techniques, we omit most of the details from this paper (detailed proofs can be found in =-=[3]-=-). This has the benefit of allowing us to swiftly proceed to the main contribution of the paper: an alternative approach to inductive proof based on infinitary reasoning with inductively defined predi... |

1 |
inductive reasoning in the logic of bunched implications
- Formalised
- 2007
(Show Context)
Citation Context ...emingly generalising the heuristic conditions applied in practice. The applicability of cyclic reasoning in FOLID has been shown by the first author to transfer to the setting of O’Hearn and Pym’s BI =-=[4]-=-, and subsequently mooted as a technique for provingSEQUENT CALCULI FOR INDUCTION AND INFINITE DESCENT 25 termination of imperative programs using separation logic [5]. Plausibly, cyclic reasoning is... |

1 |
Inductive definitions: automation and application, Higher order logic theorem proving and its applications
- Harrison
- 1995
(Show Context)
Citation Context ...al component of proof assistants and theorem provers. Often, libraries are provided containing collections of useful induction principles associated with a given set of inductive definitions, see e.g.=-=[22, 13, 24]-=-. In other cases, mechanisms permitting “cyclic” proof arguments are used, with intricate conditions imposed to ensure soundness, see e.g. [32, 23, 12]. These conditions can be broadly construed as ve... |

1 |
Descente infinie + Deduction. Logic
- Wirth
- 2004
(Show Context)
Citation Context ...th a given set of inductive definitions, see e.g.[22, 13, 24]. In other cases, mechanisms permitting “cyclic” proof arguments are used, with intricate conditions imposed to ensure soundness, see e.g. =-=[32, 23, 12]-=-. These conditions can be broadly construed as versions of the well-known mathematical principle of infinite descent originally formalised by Fermat [17]. In this article we develop proof-theoretic fo... |

1 |
General models and completeness of first-order modal µ-calculus
- Kashima, Okamoto
(Show Context)
Citation Context ..., the completeness of Kozen’s axiomatization is established relative to modal µ-frames, and our Theorem 3.6 can be seen as analogous to these results. 2 Independently to our work, Kashima and Okamoto =-=[18]-=- have extended the completeness results of [2, 16] to a first-order setting, using a notion of general model, which plays a role for the firstorder modal µ-calculus identical to that played by Henkin ... |