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A Proof Theory for Generic Judgments
, 2003
"... this paper, we do this by adding the #quantifier: its role will be to declare variables to be new and of local scope. The syntax of the formula # x.B is like that for the universal and existential quantifiers. Following Church's Simple Theory of Types [Church 1940], formulas are given the ..."
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this paper, we do this by adding the #quantifier: its role will be to declare variables to be new and of local scope. The syntax of the formula # x.B is like that for the universal and existential quantifiers. Following Church's Simple Theory of Types [Church 1940], formulas are given the type o, and for all types # not containing o, # is a constant of type (# o) o. The expression # #x.B is ACM Transactions on Computational Logic, Vol. V, No. N, October 2003. 4 usually abbreviated as simply # x.B or as if the type information is either simple to infer or not important
Higherorder abstract syntax in Coq
, 1995
"... The terms of the simplytyped λcalculus can be used to express the higherorder abstract syntax of objects such as logical formulas, proofs, and programs. Support for the manipulation of such objects is provided in several programming languages (e.g. λProlog, Elf). Such languages also provide embed ..."
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The terms of the simplytyped λcalculus can be used to express the higherorder abstract syntax of objects such as logical formulas, proofs, and programs. Support for the manipulation of such objects is provided in several programming languages (e.g. λProlog, Elf). Such languages also provide embedded implication, a tool which is widely used for expressing hypothetical judgments in natural deduction. In this paper, we show how a restricted form of secondorder syntax and embedded implication can be used together with induction in the Coq Proof Development system. We specify typing rules and evaluation for a simple functional language containing only function abstraction and application, and we fully formalize a proof of type soundness in the system. One difficulty we encountered is that expressing the higherorder syntax of an objectlanguage as an inductive type in Coq generates a class of terms that contains more than just those that directly represent objects in the language. We ove...
Automated Theorem Proving in a Simple MetaLogic for LF
 PROCEEDINGS OF THE 15TH INTERNATIONAL CONFERENCE ON AUTOMATED DEDUCTION (CADE15
, 1998
"... Higherorder representation techniques allow elegant encodings of logics and programming languages in the logical framework LF, but unfortunately they are fundamentally incompatible with induction principles needed to reason about them. In this paper we develop a metalogic M_2 which allows i ..."
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Cited by 43 (16 self)
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Higherorder representation techniques allow elegant encodings of logics and programming languages in the logical framework LF, but unfortunately they are fundamentally incompatible with induction principles needed to reason about them. In this paper we develop a metalogic M_2 which allows inductive reasoning over LF encodings, and describe its implementation in Twelf, a specialpurpose automated theorem prover for properties of logics and programming languages. We have used Twelf to automatically prove a number of nontrivial theorems, including type preservation for MiniML and the deduction theorem for intuitionistic propositional logic.
Mode and Termination Checking for HigherOrder Logic Programs
 In Hanne Riis Nielson, editor, Proceedings of the European Symposium on Programming
, 1996
"... . We consider how mode (such as input and output) and termination properties of typed higherorder constraint logic programming languages may be declared and checked effectively. The systems that we present have been validated through an implementation and numerous case studies. 1 Introduction Jus ..."
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. We consider how mode (such as input and output) and termination properties of typed higherorder constraint logic programming languages may be declared and checked effectively. The systems that we present have been validated through an implementation and numerous case studies. 1 Introduction Just like other paradigms logic programming benefits tremendously from types. Perhaps most importantly, types allow the early detection of errors when a program is checked against a type specification. With some notable exceptions most type systems proposed for logic programming languages to date (see [18]) are concerned with the declarative semantics of programs, for example, in terms of manysorted, ordersorted, or higherorder logic. Operational properties of logic programs which are vital for their correctness can thus neither be expressed nor checked and errors will remain undetected. In this paper we consider how the declaration and checking of mode (such as input and output) and termina...
A Proof of the ChurchRosser Theorem and its Representation in a Logical Framework
, 1992
"... We give a detailed, informal proof of the ChurchRosser property for the untyped lambdacalculus and show its representation in LF. The proof is due to Tait and MartinLöf and is based on the notion of parallel reduction. The representation employs higherorder abstract syntax and the judgmentsast ..."
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We give a detailed, informal proof of the ChurchRosser property for the untyped lambdacalculus and show its representation in LF. The proof is due to Tait and MartinLöf and is based on the notion of parallel reduction. The representation employs higherorder abstract syntax and the judgmentsastypes principle and takes advantage of term reconstruction as it is provided in the Elf implementation of LF. Proofs of metatheorems are represented as higherlevel judgments which relate sequences of reductions and conversions.
An Empirical Study of the Runtime Behavior of HigherOrder Logic Programs
 University of Pennsylvania. Available as
, 1992
"... this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. Government. ..."
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Cited by 29 (7 self)
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this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. Government.
The Occurrence of Continuation Parameters in CPS Terms
, 1995
"... We prove an occurrence property about formal parameters of continuations in ContinuationPassing Style (CPS) terms that have been automatically produced by CPS transformation of pure, callbyvalue terms. Essentially, parameters of continuations obey a stacklike discipline. This property was intro ..."
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We prove an occurrence property about formal parameters of continuations in ContinuationPassing Style (CPS) terms that have been automatically produced by CPS transformation of pure, callbyvalue terms. Essentially, parameters of continuations obey a stacklike discipline. This property was introduced, but not formally proven, in an earlier work on the DirectStyle transformation (the inverse of the CPS transformation). The proof has been implemented in Elf, a constraint logic programming language based on the logical framework LF. In fact, it was the implementation that inspired the proof. Thus this note also presents a case study of machineassisted proof discovery. All the programs are available in ( ftp.daimi.aau.dk:pub/danvy/Programs/danvypfenningElf93.tar.gz ftp.cs.cmu.edu:user/fp/papers/cpsocc95.tar.gz Most of the research reported here was carried out while the first author visited Carnegie Mellon University in the Spring of 1993. Current address: Olivier Danvy, Ny Munkeg...
Higherorder abstract syntax with induction in Coq
, 1994
"... Three important properties of HigherOrder Abstract Syntax are the (higherorder) induction principle, which allows proofs by induction, the (higherorder) injection principle, which asserts that equal terms have equal heads and equal sons, and the extensionality principle, which asserts that functi ..."
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Three important properties of HigherOrder Abstract Syntax are the (higherorder) induction principle, which allows proofs by induction, the (higherorder) injection principle, which asserts that equal terms have equal heads and equal sons, and the extensionality principle, which asserts that functional terms which are pointwise equal are equal. Higherorder abstract syntax is implemented for instance in the Edinburgh Logical Framework and the above principles are satisfied by this implementation. But although they can be proved at the meta level, they cannot be proved at the object level and furthermore, it is not so easy to know how to formulate them in a simple way at the object level. We explain here how SecondOrder Abstract Syntax can be implemented in a more powerful type system (Coq) in such a way as to make available or provable (at the object level) the corresponding induction, injection and extensionality principles.
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
MultiLevel MetaReasoning with Higher Order Abstract Syntax
 Foundations of Software Science and Computation Structures, volume 2620 of Lecture Notes in Computer Science
, 2003
"... Abstract. Combining Higher Order Abstract Syntax (HOAS) and (co)induction is well known to be problematic. In previous work [1] we have described the implementation of a tool called Hybrid, within Isabelle HOL, which allows object logics to be represented using HOAS, and reasoned about using tactica ..."
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Abstract. Combining Higher Order Abstract Syntax (HOAS) and (co)induction is well known to be problematic. In previous work [1] we have described the implementation of a tool called Hybrid, within Isabelle HOL, which allows object logics to be represented using HOAS, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. In this paper we describe how to use it in a multilevel reasoning fashion, similar in spirit to other metalogics such F Oλ ∆IN and Twelf. By explicitly referencing provability, we solve the problem of reasoning by (co)induction in presence of nonstratifiable hypothetical judgments, which allow very elegant and succinct specifications. We demonstrate the method by formally verifying the correctness of a compiler for (a fragment) of MiniML, following [10]. To further exhibit the flexibility of our system, we modify the target language with a notion of nonwellfounded closure, inspired by Milner & Tofte [19] and formally verify via coinduction a subject reduction theorem for this modified language. 1