Substitution Lemma Research Articles

Formal metatheory of second-order abstract syntax

Despite extensive research both on the theoretical and practical fronts, formalising, reasoning about, and implementing languages with variable binding is still a daunting endeavour – repetitive boilerplate and the overly complicated metatheory of capture-avoiding substitution often get in the way of progressing on to the actually interesting properties of a language. Existing developments offer some relief, however at the expense of inconvenient and error-prone term encodings and lack of formal foundations. We present a mathematically-inspired language-formalisation framework implemented in Agda. The system translates the description of a syntax signature with variable-binding operators into an intrinsically-encoded, inductive data type equipped with syntactic operations such as weakening and substitution, along with their correctness properties. The generated metatheory further incorporates metavariables and their associated operation of metasubstitution, which enables second-order equational/rewriting reasoning. The underlying mathematical foundation of the framework – initial algebra semantics – derives compositional interpretations of languages into their models satisfying the semantic substitution lemma by construction.

Formalisation in Constructive Type Theory of Barendregt's Variable Convention for Generic Structures with Binders

We introduce a universe of regular datatypes with variable binding information, for which we define generic formation and elimination (i.e. induction /recursion) operators. We then define a generic alpha-equivalence relation over the types of the universe based on name-swapping, and derive iteration and induction principles which work modulo alpha-conversion capturing Barendregt's Variable Convention. We instantiate the resulting framework so as to obtain the Lambda Calculus and System F, for which we derive substitution operations and substitution lemmas for alpha-conversion and substitution composition. The whole work is carried out in Constructive Type Theory and machine-checked by the system Agda.

Formalization of the Resolution Calculus for First-Order Logic

A formalization in Isabelle/HOL of the resolution calculus for first-order logic is presented. Its soundness and completeness are formally proven using the substitution lemma, semantic trees, Herbrand’s theorem, and the lifting lemma. In contrast to previous formalizations of resolution, it considers first-order logic with full first-order terms, instead of the propositional case.

LMI-based adaptive reliable H∞ static output feedback control against switched actuator failures

ABSTRACTThis paper investigates the H∞ static output feedback (SOF) control problem for switched linear system under arbitrary switching, where the actuator failure models are considered to depend on switching signal. An active reliable control scheme is developed by combination of linear matrix inequality (LMI) method and adaptive mechanism. First, by exploiting variable substitution and Finsler's lemma, new LMI conditions are given for designing the SOF controller. Compared to the existing results, the proposed design conditions are more relaxed and can be applied to a wider class of no-fault linear systems. Then a novel adaptive mechanism is established, where the inverses of switched failure scaling factors are estimated online to accommodate the effects of actuator failure on systems. Two main difficulties arise: first is how to design the switched adaptive laws to prevent the missing of estimating information due to switching; second is how to construct a common Lyapunov function based on a switched estimate error term. It is shown that the new method can give less conservative results than that for the traditional control design with fixed gain matrices. Finally, simulation results on the HiMAT aircraft are given to show the effectiveness of the proposed approaches.

Alpha-Structural Induction and Recursion for the Lambda Calculus in Constructive Type Theory

We formulate principles of induction and recursion for a variant of lambda calculus in its original syntax (i.e., with only one sort of names) where α-conversion is based upon name swapping as in nominal abstract syntax. The principles allow to work modulo α-conversion and implement the Barendregt variable convention. We derive them all from the simple structural induction principle on concrete terms and work out applications to some fundamental meta-theoretical results, such as the substitution lemma for α-conversion and the lemma on substitution composition. The whole work is implemented in Agda.

ON THE SUBSTITUTION THEOREM FOR RINGS OF SEMIALGEBRAIC FUNCTIONS

Let$R\subset F$be an extension of real closed fields, and let${\mathcal{S}}(M,R)$be the ring of (continuous) semialgebraic functions on a semialgebraic set$M\subset R^{n}$. We prove that every$R$-homomorphism${\it\varphi}:{\mathcal{S}}(M,R)\rightarrow F$is essentially the evaluation homomorphism at a certain point$p\in F^{n}$adjacent to the extended semialgebraic set$M_{F}$. This type of result is commonly known in real algebra as a substitution lemma. In the case when$M$is locally closed, the results are neat, while the non-locally closed case requires a more subtle approach and some constructions (weak continuous extension theorem, appropriate immersion of semialgebraic sets) that have interest of their own. We consider the same problem for the ring of bounded (continuous) semialgebraic functions, getting results of a different nature.

Viewing [formula omitted]-terms through maps

In this paper we introduce the notion of map, which is a notation for the set of occurrences of a symbol in a syntactic expression such as a formula or a λ term. We use binary trees over 0 and 1 as maps, but some well-formedness conditions are required. We develop a representation of lambda terms using maps. The representation is concrete (inductively definable in HOL or Constructive Type Theory) and canonical (one representative per λ term). We define substitution for our map representation, and prove the representation is in substitution preserving isomorphism with both nominal logic λ terms and de Bruijn nameless terms. These proofs are mechanically checked in Isabelle/HOL and Minlog respectively.The map representation has good properties. Substitution does not require adjustment of binding information: neither α conversion of the body being substituted into, nor de Bruijn lifting of the term being implanted. We have a natural proof of the substitution lemma of λ calculus that requires no fresh names, or index manipulation.Using the notion of map we study conventional raw λ syntax. E.g. we give, and prove correct, a decision procedure for α equivalence of raw λ terms that does not require fresh names.We conclude with a definition of β reduction for map terms, some discussion on the limitations of our current work, and suggestions for future work.

The First-Order Syntax of Variadic Functions

We extend first-order logic to include variadic function symbols and prove a substitution lemma. Two applications are given: one to bounded quantifier elimination and one to the definability of certain Borel sets.

The Algebra of Expansion

Expansion is an operation on typings pairs of type environments and result types in type systems for the λ-calculus. Expansion was originally introduced for calculating possible typings of a λ-term in systems with intersection types. This paper aims to clarify expansion and make it more accessible to a wider community by isolating the pure notion of expansion on its own, independent of type systems and types, and thereby make it easier for non-specialists to obtain type inference with flexible precision by making use of theory and techniques that were originally developed for intersection types. We redefine expansion as a simple algebra on terms with variables, substitutions, composition, and miscellaneous constructors such that the algebra satisfies 8 simple axioms and axiom schemas: the 3 standard axioms of a monoid, 4 standard axioms or axiom schemas of substitutions including one that corresponds to the usual “substitution lemma” that composes substitutions, and 1 very simple axiom schema for expansion itself. Many of the results on the algebra have also been formally checked with the Coq proof assistant. We then take System E, a λ-calculus type system with intersection types and expansion variables, and redefine it using the expansion algebra, thereby demonstrating how a single uniform notion of expansion can operate on both typings and proofs. Because we present a simplified version of System E omitting many features, this may be independently useful for those seeking an easier-to-comprehend version.

Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order Languages

Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order LanguagesFourth of a series of articles laying down the bases for classical first order model theory. This paper supplies a toolkit of constructions to work with languages and interpretations, and results relating them. The free interpretation of a language, having as a universe the set of terms of the language itself, is defined.The quotient of an interpreteation with respect to an equivalence relation is built, and shown to remain an interpretation when the relation respects it. Both the concepts of quotient and of respecting relation are defined in broadest terms, with respect to objects as general as possible.Along with the trivial symbol substitution generally defined in [11], the more complex substitution of a letter with a term is defined, basing right on the free interpretation just introduced, which is a novel approach, to the author's knowledge. A first important result shown is that the quotient operation commute in some sense with term evaluation and reassignment functors, both introduced in [13] (theorem 3, theorem 15). A second result proved is substitution lemma (theorem 10, corresponding to III.8.3 of [15]). This will be vital for proving satisfiability

Final coalgebras as greatest fixed points in ZF set theory

A special final coalgebra theorem, in the style of Aczel (1988), is proved within standard Zermelo–Fraenkel set theory. Aczel's Anti-Foundation Axiom is replaced by a variant definition of function that admits non-well-founded constructions. Variant ordered pairs and tuples, of possibly infinite length, are special cases of variant functions. Analogues of Aczel's solution and substitution lemmas are proved in the style of Rutten and Turi (1993). The approach is less general than Aczel's, but the treatment of non-well-founded objects is simple and concrete. The final coalgebra of a functor is its greatest fixedpoint.Compared with previous work (Paulson, 1995a), iterated substitutions and solutions are considered, as well as final coalgebras defined with respect to parameters. The disjoint sum construction is replaced by a smoother treatment of urelements that simplifies many of the derivations.The theory facilitates machine implementation of recursive definitions by letting both inductive and coinductive definitions be represented as fixed points. It has already been applied to the theorem prover Isabelle (Paulson, 1994).

A proof of the substitution lemma in de Bruijn's notation

This paper proves the substitution lemma using de Bruijn’s notation [3], DBN for short. DBN, which is an alternative representation of the Acalculus, represents variables with non-negative integers in order to avoid the need for cz-conversion. The substitution lemma in DBN is a wellknown result. For example, Shanker [4] formulated it in order to apply a theorem prover to the Church-Rosser Theorem, in which the substitution lemma plays an important role. However, Shanker did not give the proof, and this inspired us to prove the lemma as an exercise. We will adopt Curien’s definition of substitution [2] in DBN instead of Shanker’s one, because in Curien’s definition of substitution the treatment of terms to be substituted is encapsulated within the base step of the substitution, while in Shanker’s one it appears both in the base step and in the induction step.

Substitution revisited

A definition of simultaneous substitution for the lambda calculus is presented that is easier to work with than standard single substitution because it is a definition by structural recursion, instead of recursion on the length of terms, and bound variables are always renamed. As a result, many proofs involving substitution are by structural induction, instead of induction on the length of terms, and are simpler than the corresponding standard proofs because of the reduction in the number of cases that must be considered. Furthermore, because of the uniform renaming of bound variables, identity substitutions normalize terms with respect to equivalence up to the renaming of bound variables (α-congruence), allowing induction-free proofs of some theorems that ordinarily would be proved by induction on the relation of α-congruence. A series of results relating simultaneous substitution and α-congruence are proved, and a simple proof of the “substitution lemma” of denotational semantics is given.