calcu16 / proof

A proof editor and storage website

README for PrOOF

An online proof editor

Goals

• Formal Proof Editor with enough strength to handle arithmetic.
• Flexible enough to use different sets of basic axioms.
• Allow Hoare logic over a turing complete programming language.

Design

The initial design is based off of metamath, although I might bite the bullet on quantifiers. Specifically I will use a pattern matching system for a rule set. However it seems likely that having free and bound variables vs. distinct and possibly non-distinct variables would be simpler from a logical standpoint, even if it requires more code. For convenience, the code will be broken up into different types: operators, properties, variables, and formulas.

Note: for the following section return etc. refers to the type of a variable.

Operators are either quantifiers or relations. A quantifier such as for all, or there exits binds a variable and makes a statement about that variable. They take in a variable name and return a formula. Relations take in formulas and return formulas.

Variables are the basic elements of the universe. In the initial axiom they will represent sets, and propertes will be used for ZFC specification and replacement.

Formulas are logical elements, in classical logic they are either true or false for a given universe (LEM). They can take arguments to signal unification.

For instance we can create the following axioms for propositional calculus

        |- a -> (b -> a)
        |- (a -> (b -> c)) -> (a -> b) -> (a -> c)
        |- (!a -> !b) -> (b -> a)
a, a->b |- b