James McKinna and Robert Pollack. Some lambda calculus and type theory formalized. Journal of Automated Reasoning, 23(3--4), November 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....without an explicit axiomatization. Usually only a short definition is given in natural language, followed by some examples [1] Actually, # equivalence can be defined rigorously in more than one way; for instance, besides the conventional one [1] there is the variant used by McKinna Pollack [15], and Gabbay Pitts alternative, based on the notion of variable transposition [6, 7] It is therefore natural to show formally that these three definitions are really equivalent. Research supported by the MURST Project tosca. This is a preliminary version. The final version will be published ....

.... [1] 2 First Order Abstract Syntax approach As a first case study, we will address the problem of encoding three notions of # equivalence: the conventional one given in common textbooks on # calculi (see, e.g. 1] a variant used in a previous formal study of the metatheory of # calculus [15], and an alternative formulation proposed in [6] We are not interested here in taking a HOAS based approach in encoding the syntax of the object language. Otherwise, # equivalence would be provided for free by the metalanguage of the logical framework and would not be accessible at the object ....

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J. McKinna and R. Pollack. Some lambda calculus and type theory formalized. Journal of Automated Reasoning, 23(3-4), Nov. 1999.

....are inductive, and packages to reason about such constructions are by now fairly common currency among theorem provers. One particularly interesting aspect is that the nu calculus uses name abstraction as well as lambda abstraction. Reasoning about the latter is still a delicate area see [5, 11, 18] for some approaches and concentrating attention onto pure names may provide some useful insights. Note that we are not concerned here with an implementation of the proof that the reasoning system itself is correct (Theorem 9) what might benefit from machine assistance is the demonstration ....

J. McKinna and R. Pollack. Some lambda calculus and type theory formalized. Submitted for publication, 1997.

....Cons(i; b) env s : a A straightforward proof of this by induction on the derivation of the typing of s runs into problems in choosing suitably fresh names for the variables introduced in the abstraction case. A proof technique which overcomes these diculties is given by McKinna and Pollack [20]. The trick is to de ne a second version of the typing judgement in which the introduction rule for an abstraction has a universal quanti er 8n : newIn n env newFor n s Cons(n; a) env inst 0 (Var n) s : b dbtype abs 0 env (Abs s) a b and the reader should compare this rule to ....

J. McKinna and R. Pollack. Some lambda calculus and type theory formalized. JAR, 1998.

....Cons(i; b) env s : a A straightforward proof of this by induction on the derivation of the typing of s runs into problems in choosing suitably fresh names for the variables introduced in the abstraction case. A proof technique which overcomes these diculties is given by McKinna and Pollack [21]. The trick is to de ne a second version of the typing judgement in which the introduction rule for an abstraction has a universal quanti er 8n : newIn n env newFor n s Cons(n; a) env inst 0 (Var n) s : b dbtype abs 0 env (Abs s) a b and the reader should compare this rule to ....

.... Progress Lemma uses the standard typing judgement is sucient and does not need totality of substitution. 5 Related Work In the literature, there are several large scale machine assisted proofs of properties of languages with variable binding using rst order encodings, see for example [34, 15, 21, 3]. Here we review papers that try to overcome the problems of rst order encodings by using some form of HOAS; there are other intermediate approaches aimed at reducing the aforementioned issue [33, 12] which we do not have the space here to mention. We can distinguish two main (and not ....

J. McKinna and R. Pollack. Some lambda calculus and type theory formalized. JAR, 1998.

....of the pi calculus by a number of researchers, including the present paper. The parameters are: the theorem proving system used; the style of pi calculus (monadic or polyadic) the approach to binding (names, de Bruijn indices [1] higher order abstract syntax (HOAS) 19] McKinna and Pollack s [13] approach (MP) Gabbay and Pitts [4] FM sets approach (GP) the style of operational semantics (labelled transition system (LTS) or reduction relation (RED) the focus on bisimulation ( or typing ( Author Prover Calculus Binding Semantics Focus Melham [14] HOL monadic names LTS ....

J. McKinna and R. Pollack. Some lambda calculus and type theory formalized. Journal of Automated Reasoning, 23(3), 1999.

....be proven on the term structure. They can then be lifted to the algebra level. Reasoning about imperative and concurrent programs works in a similar way [4,14,15] The Church Rosser theorem in the calculus has been considered interesting for interactive proof checkers by many researchers (c.f. [21,17,18] and the references given there. Almost all previous attempts formally reconstruct a proof via the nowadays standard methods of Tait MartinL of (c.f. 2] or Takahashi [26] They use induction and therefore higher order logic. Our proofs are rst order and often nite combinatorics. We ....

....the abstract Church Rosser result of corollary 2, which itself is a corollary of proposition 1. 2 8 More Church Rosser Calculations A main application of our results is mechanized proof checking. This has already been done for the Church Rosser theorem by various authors and methods; notice only [21,17,18]. Most of these implementations do not follow our proof from the previous chapter, but the nowadays standard methods of Tait Martin L of (c.f. 2] and Takahashi [26] Our proof uses more diagrammatics, the standard one does more work at the term level. The standard proof is shorter, whereas ....

J. McKinna and R. Pollack. Some lambda calculus and type theory formalized. J. Automated Reasoning, 23(3-4):373-409, 99.

....a definition can then be used to implement # conversion, so that a proper notion of substitution can take care of name capture, usually applying normalization. A theory of # equivalence can be obtained along the same lines as done for approach (2) in 9 straightforward permutations [8, 9] PTS [19] deBruijn [4] method simple renaming, substitution on top permutations parameters vs. variables, two substitutions nameless variables ## # equivalence, # conversion, # reduction # equivalence, # conversion # equivalence, # conversion, # reduction # equivalence is identity, ....

J. McKinna and R. Pollack. Some lambda calculus and type theory formalized. Journal of Automated Reasoning, 23(3--4):373--409, 1999.

....of the pi calculus by a number of researchers, including the present paper. The parameters are: the theorem proving system used; the style of pi calculus (monadic or polyadic) the approach to binding (names, de Bruijn indices, higher order abstract syntax (HOAS) McKinna and Pollack s [9] approach (MP) the style of operational semantics (labelled transition system (LTS) or reduction relation (RED) the focus on bisimulation ( or typing ( Author Prover Calculus Binding Semantics Focus Melham [10] HOL monadic names LTS Hirschko [5] Coq polyadic de Bruijn LTS Honsell ....

J. McKinna and R. Pollack. Some lambda calculus and type theory formalized. Journal of Automated Reasoning, 23(3), 1999.

....nature of the problems are. Freely assuming # equality has by now become standard although it is highly informal and not immediately formalisable. In fact, a number of publications have recently been devoted to the presentation of ways of formalising # equality and higher order languages, e.g. [12, 15, 16, 11, 20]. All of these distance 1 Observe that [4] also deals with rewriting and NbE, although in a di#erent sense: they present the NbE algorithm relative to a domain theoretic model of a certain class of rewriting systems. 2 Types # : a # # # #a.# FTV(a) a FTV(# 1 # # 2 ) ....

James McKinna and Randy Pollack. Some lambda calculus and type theory formalized. To appear in Journal of Automated Reasoning.

.... UTT (see eg, 15] Typed operational semantics [7, 8] Coercive subtyping [16, 29] cf, implementation of coercions in Lego [2] and in Coq [27] Extensionality and related issues [9] ffl Application examples: Formalisation of Pure Type Systems [3] and verification of proof checker [24, 22, 25] Verification of functional programs [4] imperative programs [11, 28] and concurrent programs [30] Model checking in Lego [31] SN proof of the system F [1] Formalisation of synthetic domain theory in Lego [26] ffl Development of representation schemes methods: Program ....

J. H. McKinna and R. Pollack. Some lambda calculus and type theory formalized. Journal of Automated Reasoning, 1999. Forthcoming.

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J. cKinna and R. Pollack . Some lambda calculus and type theory formalized. Journal of Automated Reasoning, 23(3--4), November 1999.

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James McKinna and Robert Pollack. Some lambda calculus and type theory formalized. Journal of Automated Reasoning, 23(3--4), November 1999.

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J. McKinna and R. Pollack. Some lambda calculus and type theory formalized. Journal of Automated Reasoning, 23(3--4), Nov. 1999.

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James McKinna and Robert Pollack. Some lambda calculus and type theory formalized. J. Autom. Reason., 23(3):373--409, 1999.

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Robert Pollack James McKinna. Some lambda calculus and type theory formalized, 1997.

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J. McKinna and R. Pollack. Some lambda calculus and type theory formalized. Journal of Automated Reasoning, 23(3--4), November 1999.

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James McKinna and Robert Pollack. Some lambda calculus and type theory formalized. JAR, 1998.

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