N. Shankar. A mechanical proof of the Church-Rosser theorem. Journal of the Association for Computing Machinery, 35(3):475--522, July 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....glitches and suggesting how to fix them. Also, a formal development is easy to come back to a year later, as all the details you would not otherwise have written down are explicit, and don t have to be rediscovered. 1. 2 Related Work There are many formalizations of the Church Rosser theorem [Sha85, Hue94, Nip96, Pfe92] the only formalization of a standardization theorem we know of is [Coq96a] for lazy combinator expressions. Formalizations of type theory include [DB93, Bar96] both of these address limited aspects of very special type theories (essentially the Calculus of Constructions) ....

....feel needs checking. For example, both authors have checked parts of type theory papers we were asked to referee. A novelty in our presentation is the use of named variables. Most of the formalizations of type theory or lambda calculus that we know of use de Bruijn indices ( nameless variables ) Sha85, Alt93, Hue94, Nip96, Bar96] or higher order abstract syntax [Pfe92] to avoid formalizing the renaming of variables to prevent unintended capture during substitution. While de Bruijn notation is concrete and suitable for formalization, there are reasons to formalize the theory with named ....

N. Shankar. A mechanical proof of the Church-Rosser theorem. Technical Report 45, Institute for Computing Science, University of Texas at Austin, March 1985.

....Tait Martin L of simpli cation and the Takahashi Triangle. A classic presentation may be found in Barendregt [1] The proofs involve sophisticated inductive arguments, whose patterns have also intrigued researchers in mechanically checked proof. The rst mechanical proof was presented by Shankar [9], and has been followed by Huet [5] Nipkow [7] Pfenning [8] Vestergaard Brotherston [11] and Ford Mason [2] Of these, only Nipkow extends the work beyond reduction to proofs of con uence for and reduction. One key issue in these proofs is whether the syntax of the lambda calculus ....

....as in [2] we wish to address the issues of name carrying syntax, in order to relate more directly to practical programming languages. The presence of names raises as key issues the de nitions of equivalence, substitution, and reduction. In two of the above proofs where names were treated [9, 11], con uence was proved for an arbitrary intermixture of and reduction. This intermixture bred an unfortunate complexity. 2 P. V. Homeier The elegant presentation by Barendregt [1] axes this complexity by the Barendregt Variable Convention (BVC) Our aim was to follow Barendregt as closely ....

Shankar, N.: A Mechanical Proof of the Church-Rosser Theorem, Journal of the ACM Vol. 35, No. 3, July 1988, 475-522.

....of these entities, for example the wellformedness of a formula or the validity of a proof. However, reasoning about languages with binders in a first order representation is a tedious business and requires numerous technical lemmata concerning variable renaming, substitution etc. see e.g. [Sha88], Alt93] More suitable is a logical framework with a higher order term language which has binders itself. Then object variables can be encoded by meta variables and substitution in the object theory can be expressed by substitution in the logical framework. The primary candidate for a ....

Natarajan Shankar. A mechanical proof of the Church-Rosser theorem. Journal of the ACM, 35(3):475--522, July 1988.

....for this proof. Our verification of the critical pair lemma required several inductions w.r.t. functions like rewrites whose domains are undecidable. Thus, our proof differs substantially from other case studies in related areas (e.g. the proofs of the Church Rosser theorem for the calculus in [16, 19]) In Section 2 we introduce the data types and in Section 3 we give the definitions of all algorithms used. The remaining sections contain theorems proved with our calculus. For a detailed overview the reader is referred to the table of contents at the end of the report. Several of these ....

N. Shankar. A Mechanical Proof of the Church-Rosser Theorem, Journal of the ACM, 35(3):475-522, 1988.

....future evolution. 1 Introduction In the last decades, a large variety of provers 1 implementing various kinds of reasoning have been developed and used to accomplish complex tasks. For instance, interactive theorem provers have been employed to mechanically prove deep mathematical theorems ([17,35,37,7]) as well as correctness of nontrivial hardware and software systems ( 6,8,36,44] As another example, the use of model checking tools has become a de facto standard approach (also in many industrial projects) to validate the design of hardware components or communication protocols ( 11,24] ....

....( 26] is a state of the art inductionbased theorem prover which extends and enhances the Boyer Moore prover, Nqthm. acl2 embeds extremely powerful heuristics and decision procedures; it has been successfully applied to prove mathematical theorems, and to verify properties of systems (see, e.g. [6,8,37]) In acl2, the proof of a conjecture is driven by a number of heuristics, called processes, integrating several inference techniques (such as simpli cation by a database of rewrite rules, elimination of irrelevants, cross fertilization, etc. These processes are called within a toplevel search ....

N. Shankar. A mechanical proof of the Church-Rosser theorem. JACM, 35(3):475-522, 1988.

....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 ....

N. Shankar. A mechanical proof of the Church-Rosser theorem. Journal of the ACM, 35(3):475-522, 1988.

....glitches and suggesting how to fix them. Also, a formal development is easy to come back to a year later, as all the details you would not otherwise have written down are explicit, and don t have to be rediscovered. 1.2. Related work There are many formalizations of the Church Rosser theorem [44, 20, 32, 33]; the only formalization of a standardization theorem we know of is for lazy combinator expressions [8] Formalizations of type theory include [11, 4] both of these address limited aspects of very special type theories (essentially the Calculus of Constructions) although [4] is very interesting ....

....feel needs checking. For example, both authors have checked parts of type theory papers we were asked to referee. A novelty in our presentation is the use of named variables. Most of the formalizations of type theory or lambda calculus that we know of use de Bruijn indices ( nameless variables ) [44, 1, 20, 32, 4] or higher order abstract syntax [33] to avoid formalizing the renaming of variables to prevent unintended capture during substitution. While de Bruijn notation is concrete and suitable for formalization, there are reasons to formalize the theory with named variables. For one thing, ....

Shankar, N.: 1988, `A Mechanical Proof of the Church-Rosser Theorem'. Journal of the ACM 35(3), 475--522.

....done by the human. For further details and for a collection of numerous other theorems about partial functions proved with our calculus see [27] 12 Thus, our proof differs substantially from other case studies in related areas (e.g. the proofs of the Church Rosser theorem for the calculus in [59, 69]) ibn 98 48.tex; 4 01 2000; 17:21; no v. p.38 INDUCTION PROOFS WITH PARTIAL FUNCTIONS 39 Note that apart from reasoning about given partial functions, our approach is also required for program schemes where termination of the program depends on the instantiation of the auxiliary functions which ....

Shankar, N., `A Mechanical Proof of the Church-Rosser Theorem', Journal of the ACM 35, 475-522 (1988).

.... interpreted as the set of all possible complete and consistent extensions, cf. also [WG94, Type D 0 ] This corresponds to the intuition that such 3 In that respect, our proofs differ from other case studies in related areas (e.g. the proofs of the Church Rosser theorem for the calculus in [Sha88, Nip96] 9 a function is not really partial, but it is a total function with (partly) unknown behaviour. Hence, this approach cannot be used for non terminating functions like f(x) s(f(x) which do not have a complete consistent extension. In contrast, in our approach every specification is ....

Shankar, N., A Mechanical Proof of the Church-Rosser Theorem, Journal of the ACM 35, 475-522, 1988.

.... calculus. For example, Huet [5] in the Coq system or Nipkow [11] in Isabelle HOL. A comparison is dicult because our goal was di erent and, more important, the logics involved are signi cantly di erent: ACL2 logic is a much weaker logic than those of Coq or HOL. A more related work is Shankar [13], using Nqthm. Although his work is on concrete reduction relations from the calculus and he does not deal with the abstract case, some of his ideas are re ected in our work. To our knowledge, no formalization of term rewriting systems has been done yet and consequently the formal proofs of ....

N. Shankar. A mechanical proof of the Church-Rosser theorem. Journal of the ACM, 35(3):475-522, 1988.

....the root. 1 2 3 1 . PI n878 12 Solange COUPET GRIMAL et Olivier RIDOUX Our representation is more suitable for Logic Programming because the renaming problem raised by fi reductions is not handled by counting the nodes labelled lambda, as it is the case with De Bruijn s notation (see [53] for general algorithms for this notation) The renaming problem is solved by the unification of rational terms, which is automatically performed by Prolog II. Thus, the resulting normalization algorithm 9 is particularly concise. The missing definitions like member already belong to any Logic ....

N. Shankar. A Mechanical Proof of the Church-Rosser Theorem. Technical Report 78712, Institute for Computing Science, The University of Texas at Austin, 1985.

....ith on the path from this occurrence to the root: 14 . 1 2 3 1 Our representation is more suitable for Logic Programming because the renaming problem raised by fi reductions does not need to be handled by counting the nodes labeled lambda as is the case with de Bruijn s notation (see [51] for general algorithms for this notation) The renaming problem is solved by the unification of rational terms, which is automatically performed by Prolog II. Thus, the resulting normalization algorithm 9 is particularly concise. The missing definitions like member already belong to any Logic ....

N. Shankar. A Mechanical Proof of the Church-Rosser Theorem. Technical Report 78712, Institute for Computing Science, The University of Texas at Austin, 1985.

....package [Pau94a] on a large proof development, definition of recursive functions, and how the automation tools of Isabelle can be employed effectively to solve major proof steps. Formalisation of this proof is not new and the first mechanical proof was done in the Boyer Moore prover by Shankar [Sha88] and has later been described by Huet as mentioned above and in for example [Pfe92] The Church Rosser theorem has also been proved for combinatorial logic in for example [CM92] and by Paulson contained in the Isabelle ZF distribution. The proofs for combinatorial logic are however much simpler. ....

....else Var(i) substrec(u,Fun(t) k) Fun(substrec(u,t,succ(k) substrec(u,App(b,f,a) k) App(b,substrec(u,f,p) substrec(u,a,p) This requires that the lift function takes an extra argument, k, to lift a term k steps by use of addition. Our formulation is similar to the one used by Shankar [Sha88] and completely eliminates the use of addition thus saving a large number of lemmas. The size of the substitution proofs decreased by over 1 3 after changing the definitions to the current formulation, thus computational considerations can be expensive in proofs. However, as the substitution ....

N. Shankar. A mechanical proof of the Church-Rosser theorem. Journal of the Association for Computing Machinery, 35(3):475--522, July 1988.

....lemmas and the order of their presentation, a user may guide the theorem prover through the verification of complicated theorems. The Boyer Moore logic and prover have been used to specify and verify numerous difficult problems including Goedel s Incompleteness Theorem and the Church Rosser Theorem[26, 25], a microprocessor[13] an assembler[22] and a compiler[31] The latter three proofs have been formally integrated into a stack of machines[23, 1] 2.2.1 The Nqthm Logic The Nqthm version of the Boyer Moore logic is a quantifier free first order logic with equality that permits recursive ....

N. Shankar. A mechanical proof of the church-rosser theorem. Journal of the ACM, 35:475--522, 1988.

.... the integers (Bevier, Kaufmann, and Wilding, Bev88] numbers naturals.events ) a library of useful definitions and lemmas about the natural numbers (Wilding, numbers nim.events ) a formalization of the game of Nim and a proof that a certain algorithm implements a winning strategy (Shankar, [Sha88], shankar church rosser.events ) a proof of the Church Rosser theorem for lambda calculus (Shankar, Sha86] shankar goedel.events ) a proof of Godel s incompleteness theorem for Shoenfield s first order logic extended with Cohen s axioms for hereditarily finite set theory, Z2 (Shankar, ....

N. Shankar. A mechanical proof of the Church-Rosser theorem. JACM, 35(3):475--522, 1988.

.... a capability that has often been exploited to use Nqthm to do proofs in other computational logics: Nqthm has proved the soundness and completeness of a propositional calculus decision procedure [8] the Turing completeness of pure Lisp [10] the Church Rosser theorem for lambda calculus [33], and the soundness of the proof rules of Misra and Chandy s Unity logic [14, 16] ffl System Verification Perhaps most representative of digital systems verification is the now classic example, the CLI short stack, which combines both hardware and software verification. The short stack ....

N. Shankar. A Mechanical Proof of the Church-Rosser Theorem. JACM 35(3), 475--522, 1988. Design Goals for ACL2 CLI Technical Report #101 41

.... their prime factors) The result is a theory that, with some support for automating theorem proving, is e ective enough to be used to prove complex metatheorems of the sort we are interested in here, including, among other things, G odel s incompleteness theorem and the Church Rosser theorem [15,16]. Unfortunately however, the Boyer Moore logic still shares the problem with arithmetic of lacking facilities that allow a user to scope theorems with contexts in the manner discussed in x2, and illustrated by the examples (1) and (2) A recent development that provides a solution to this problem ....

N. Shankar. A mechanical proof of the Church-Rosser theorem. J. Assoc. Comput. Mach., 35:475-522, 1988.

....it is a practical example of using metatheory to extend a theorem prover, but also because it works with a logic that is, in many ways, similar to FS 0 in that it resembles Lisp restricted to primitive recursive functions 3 . Another example of metatheory in Nqthm is the proof, by Shankar [21], of the Church Rosser theorem. The distinction here is that this is metatheory as mathematics for its own sake, rather than as a means for making work in the object theory easier. However it is one of the most substantial metatheoretic results that has been formalised and machine checked. Also, ....

Natarayan Shankar. A Mechanical Proof of the Church-Rosser Theorem. Journal of the ACM, 35:475--522, 1988.

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N. Shankar. A mechanical proof of the Church-Rosser theorem. Journal of the Association for Computing Machinery, 35(3):475--522, July 1988.

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N. Shankar. A mechanical proof of the Church-Rosser theorem. Journal of the ACM, 35(3):475-- 522, 1988.

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N. Shankar. A mechanical proof of the Church-Rosser theorem. Journal of the Association for Computing Machinery, 35(3):475--522, July 1988.

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N. Shankar. A mechanical proof of the Church-Rosser theorem. Journal of the ACM, 35(3):475--522, 1988.

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N. Shankar. "A mechanical proof of the Church-Rosser Theorem." Technical Report 45, Institute for Computing Science, The University of Texas at Austin, March 1985. 78

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N. Shankar. "A mechanical proof of the Church-Rosser Theorem." Technical Report 45, Institute for Computing Science, The University of Texas at Austin, March 1985. 76

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N. Shankar (1988), A mechanical proof of the Church-Rosser theorem, Journal of the Association for Computing Machinery, 35(3):475-522.

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