HOL Light QE

Abstract

We are interested in algorithms that manipulate mathematical expressions in mathematically meaningful ways. Expressions are syntactic, but most logics do not allow one to discuss syntax. cttqe is a version of Church's type theory that includes quotation and evaluation operators, akin to quote and eval in the Lisp programming language. Since the HOL logic is also a version of Church's type theory, we decided to add quotation and evaluation to HOL Light to demonstrate the implementability of cttqe and the benefits of having quotation and evaluation in a proof assistant. The resulting system is called HOL Light QE. Here we document the design of HOL Light QE and the challenges that needed to be overcome. The resulting implementation is freely available.

Towards Specifying Symbolic Computation

“…Third by formally proving in ctt uqe [12] and ctt qe [13] the mathematical meanings of SBMAs from their formal definitions. And fourth by further developing HOL Light QE [5] so that these SBMA definitions and the proofs of their mathematical meanings can be performed and machine checked in HOL Light QE. As a small startup example, we have defined a symbolic differentiation algorithm for polynomials and proved its mathematical meaning from its definition in [13, subsections 4.4 and 9.3].…”

“…(4) says that, if the input represents a rational expression in x q , then either the input and output denote the same member of f or they both do not denote any member of f (the semantic component). And (5) says that, if the input does not represent a rational expression in x q , then the output is undefined. specNormRatFun o is the formula…”

“…(3-4) say that, if the input to RatFun →o represents a rational function in x q , then the output represents a rational function in x q whose body is in quasinormal form (the syntactic component). (5) says that, if the input represents a rational function in x q , then input and output denote the same (possibly partial) function on the rational numbers (the semantic component). And (6) says that, if the input does not represent a rational function in x q , then the output is undefined.…”

“…(3) says that, if the input u e to specDiff o is a member of the subtype DiffExpr →o , then the output is also a member of DiffExpr →o (the syntactic component). (4)(5)(6) say that, if the input is a member of DiffExpr →o and, for all real numbers a, if the function f represented by u e is differentiable at a, then the derivative of f at a equals the function represented by diff → u e at a (the semantic component). And (7) says that, if the input is not a member of DiffExpr →o , then the output is undefined.…”

Incorporating quotation and evaluation into Church's type theory

“…(24) is an instance of part 5 of Theorem 9.3.1;(25) follows from (24) by Weakening;(26) and(27) follow from the hypothesis E 2 o and Axioms B5.1-6 by Universal Generalization, Universal Instantiation, the Equality Rules and propositional logic;(28) and(29)follow from (26), (27), and the hypothesis D o by Universal Instantiation, parts 5 and 7 of Lemma 9.3.2, the Equality Rules, and propositional logic; (30) and (31) follow from (28), (29), and the hypothesis E 2 ′ and propositional logic; (40)-(43) follow from the hypothesis E 2 o and the definition of IS-EFFECTIVE by Beta-Reduction by Substitution, part 4 of Lemma 6.6.2 and Lemma 9.3.2; and (44) follows from (41)-(43) by Lemma 6.3.7.…”