Declarative Representation of Proof Terms

Coen, Claudio Sacerdoti

doi:10.1007/s10817-009-9136-7

Cited by 11 publications

(8 citation statements)

References 10 publications

(7 reference statements)

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“…It is worth noting that in our case, the justifications appearing in a declarative proof script are CIC terms of degree 3 being fragments of the underlying proof term [13].…”

Section: Contents and Structure Of Proofsmentioning

confidence: 95%

“…To this aim, a declarative proof script contains a lot of formal propositions that, for the reader's convenience, state the premises and the conclusion of most proof steps [13][14][15]. In addition, the intended meaning of a proof step can be clarified by introducing some natural language in the concrete syntax of its representation.…”

Section: Contents and Structure Of Proofsmentioning

confidence: 99%

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Procedural Representation of CIC Proof Terms

Guidi

2009

J Autom Reasoning

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We propose an effective procedure, the first one to our knowledge, for translating a proof term of the Calculus of Inductive Constructions (CIC), into a tactical expression of the high-level specification language of a CIC-based proof assistant like coq (Coq development team 2008) or matita (Asperti et al., J Autom Reason 39: 2007). This procedure, which should not be considered definitive at its present stage, is intended for translating the logical representation of a proof coming from any source, i.e. from a digital library or from another proof development system, into an equivalent proof presented in the proof assistant's editable high-level format. To testify to effectiveness of our procedure, we report on its implementation in matita and on the translation of a significant set of proofs (Guidi, ACM Trans Comput Log 2009) from their logical representation as coq 7.3.1 (Coq development team 2002) CIC proof terms to their high-level representation as tactical expressions of matita's user interface language.

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“…It is worth noting that in our case, the justifications appearing in a declarative proof script are CIC terms of degree 3 being fragments of the underlying proof term [13].…”

Section: Contents and Structure Of Proofsmentioning

confidence: 95%

Section: Contents and Structure Of Proofsmentioning

confidence: 99%

Procedural Representation of CIC Proof Terms

Guidi

2009

J Autom Reasoning

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“…In Matita, these rendered proof objects also can be read back in, and checked for correctness like in a declarative proof system [45].…”

Section: 2mentioning

confidence: 99%

A Synthesis of the Procedural and Declarative Styles of Interactive Theorem Proving

Wiedijk¹

2012

Logical Methods in Computer Science

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Abstract. We propose a synthesis of the two proof styles of interactive theorem proving: the procedural style (where proofs are scripts of commands, like in Coq) and the declarative style (where proofs are texts in a controlled natural language, like in Isabelle/Isar). Our approach combines the advantages of the declarative style -the possibility to write formal proofs like normal mathematical text -and the procedural style -strong automation and help with shaping the proofs, including determining the statements of intermediate steps.Our approach is new, and differs significantly from the ways in which the procedural and declarative proof styles have been combined before in the Isabelle, Ssreflect and Matita systems. Our approach is generic and can be implemented on top of any procedural interactive theorem prover, regardless of its architecture and logical foundations.To show the viability of our proposed approach, we fully implemented it as a proof interface called miz3, on top of the HOL Light interactive theorem prover. The declarative language that this interface uses is a slight variant of the language of the Mizar system, and can be used for any interactive theorem prover regardless of its logical foundations. The miz3 interface allows easy access to the full set of tactics and formal libraries of HOL Light, and as such has 'industrial strength'.Our approach gives a way to automatically convert any procedural proof to a declarative counterpart, where the converted proof is similar in size to the original. As all declarative systems have essentially the same proof language, this gives a straightforward way to port proofs between interactive theorem provers.

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“…The main language used to write proof scripts is procedural, in the LCF tradition, and essentially similar to the one used by Coq. In addition to that, Matita features a declarative language ( [13]), in the style usually ascribed to Trybulec (the so called Mizar-style [19]), and made popular in the ITP community mostly by the work of Wenzel [29].…”

Section: Proof Authoringmentioning

confidence: 99%

The Matita Interactive Theorem Prover

Asperti

Ricciotti

Coen

et al. 2011

Lecture Notes in Computer Science

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Matita is an interactive theorem prover being developed by the Helm team at the University of Bologna. Its stable version 0.5.x may be downloaded at http://matita.cs.unibo.it. The tool originated in the European project MoWGLI as a set of XML-based tools aimed to provide a mathematician-friendly web-interface to repositories of formal mathematical knoweldge, supporting advanced content-based functionalities for querying, searching and browsing the library. It has since then evolved into a light but fully fledged ITP, particularly suited for the assessment of innovative ideas, both at foundational and logical level. In this paper, we give an account of the whole system, its peculiarities and its main applications. The systemMatita is an interactive proof assistant, adopting a dependent type theory -the Calculus of (Co)Inductive Constructions (CIC) -as its foundational language for describing proofs. It is thus compatible, at the proof term level, with Coq [27], and the two systems are able to check each other's proof objects. Since the two systems do not share a single line of code, but are akin to each other, it is natural to take Coq as the main term of comparison, referring to other systems (most notably, Isabelle and HOL) when some ideas or philosophies characteristic of these latter tools have been imported into our system.Similarly to Coq, Matita follows the so called De Bruijn principle, stating that proofs generated by the system should be verifiable by a small and trusted component, traditionally called kernel. Unsurprisingly, the kernel has roughly the same size in the two tools, in spite of a few differences in the encoding of terms: in particular, Matita's kernel handles explicit substitutions to mimic Coq's Section mechanism, and can cope with existential metavariables, i.e. non-linear placeholders that are Curry-Howard isomorphic to holes in the proofs. Metavariables cannot be instantiated by the kernel: they are considered as opaque constants, with a declared type, only equal to themselves.While this extension does not make the kernel sensibly more complex or fragile, it has a beneficial effect on the size of the type inference subsystem, here

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Declarative Representation of Proof Terms

Cited by 11 publications

References 10 publications

Procedural Representation of CIC Proof Terms

Procedural Representation of CIC Proof Terms

A Synthesis of the Procedural and Declarative Styles of Interactive Theorem Proving

The Matita Interactive Theorem Prover

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