Eqthy

Version 0.6 | See also: Lome ∘ Philomath ∘ LCF-style-ND

Eqthy is a formal language for equational proofs. Its design attempts to reconcile simplicity of implementation on a machine with potential for usability by a human (more on this below). It supports an elementary linear style, where each line gives a step which is derived from the step on the previous line, and may optionally state the justification for the derivation in that step.

Here is a simple example: a proof that, in a monoid, the identity element commutes with any element.

axiom (idright) mul(A, e) = A
axiom (idleft)  mul(e, A) = A
axiom (assoc)   mul(A, mul(B, C)) = mul(mul(A, B), C)
theorem (idcomm)
    mul(A, e) = mul(e, A)
proof
    A = A
    mul(A, e) = A           [by idright]
    mul(A, e) = mul(e, A)   [by idleft]
qed

For improved human usability, Eqthy is usually embedded within Markdown documents. This allows proofs to be written in a more "literate" style, with interspersed explanatory prose and references in the form of hyperlinks.

For a fuller description of the language, including a set of Falderal tests, see doc/Eqthy.md.

A number of proofs have been written in Eqthy to date. These can be found in the eg/ directory. In particular, there are worked-out proofs:

• in Group Theory, including:
  • some simple theorems about the inverse element;
  • the Socks and Shoes theorem;
• in Ring Theory;
• in Boolean Algebra, including:
  • De Morgan's laws;
  • the Absorption laws in an alternate axiomatization;
  • some properties of set difference;
• in Interior Algebra (building on Boolean algebra);
• in Propositional Algebra;
• in Johnson's 1892 axiom system for propositional logic, including:
  • the Rule of Constancy
• in Combinatory Logic;

with hopefully more to come in the future.

The Eqthy language is still at an early stage and is subject to change. However, since the idea is to accumulate a database of proofs which can be built upon, it is unlikely that the format of the language will change radically.

Design Principles

Probably the language that Eqthy most resembles, in spirit, is Metamath; but its underlying mechanics are rather different. Eqthy is based on equational logic, so each step is an equation.

Eqthy's design attempts to reconcile simplicity of implementation on a machine with human usability. It should be understood that this is a balancing act; adding features to the language which improve usability will generally be detrimental to simplicity, and vice versa.

It has been implemented in Python 3 in about 760 lines of code; the core (term manipulation and verifier modules) is about 340 lines of code. For more details, see the Implementations section below.

It is also possible for a human to write Eqthy documents by hand, and to read them, without much specialized knowledge. The base logic is equational logic, which has only 5 rules of inference, and these rules are particularly widely understood; "replace equals with equals" is a standard part of the high-school algebra cirriculum.

In comparison, mmverify.py, a Python implementation of a Metamath checker, is about 630 lines of code; and while it is undoubtedly simple, the Metamath language is not widely regarded as being easy to write or read.

In fairness it should not go unstated that equational logic, lacking as it does implication and existential quantifiers, is widely regarded as more awkward to use than first-order predicate logic as a formal language for proofs.

Implementations

While the language does not prescribe any specific application for proofs written in Eqthy, it is reasonable to expect that one of the main reasons one would want a computer to read one would be for it to check it for validity.

This distribution contains such a proof checker, written in pure Python 3, with no dependencies. It has been tested under 3.10, 3.11, and 3.14. Its source code can be found in the src/ directory.

The core module that does proof checking, eqthy.verifier, is less than 200 lines in length, despite having many logging statements (which both act as comments, and provide a trace to help the user understand the execution of the verifier on any given document).

The desire is to make reading the code and understanding its behaviour as un-intimidating as possible.

For todo items for this project, see TODO.md.