Philomath

Version 1.1 | See also: LCF-style-Natural-Deduction ∘ Maxixe ∘ Cardboard Prolog

Philomath is an LCF-style theorem prover written in ANSI C. It implements classical propositional logic inside a Natural Deduction system with labelled assumptions. For more information on this approach, see the LCF-style-Natural-Deduction article.

How do I write a proof with this?

Create a file myproof.c:

#include "formula.h"
#include "proof.h"
int main(int argc, char **argv) {
    /* Proof of p -> p */
    return proves(
        impl_intro(1, suppose(var("p"), 1)),
        impl(var("p"), var("p"))
    );
}

Then compile and run it with:

./build-proof.sh myproof

And run the resulting executable:

./myproof.exe

If the exit code is 0, the proof is valid!

% echo $?
0

If you wish to build the proof with debugging output, you can pass the -D flag:

./build-proof.sh -D myproof

Limitations

LCF-style theorem proving relies on data abstraction, i.e. information hiding. In C, information hiding is accomplished by exposing, in a header file, only the fact that a struct exists, and not exposing its structure. Client code can then work with pointers to that struct without knowing what's contained in it.

It is however possible to subvert this mechanism. If your proof code does any of the following things, it is possible it is no longer a valid proof:

• includes the file proof.c directly, rather than including proof.h
• casts a struct theorem value to some other type
• casts a value of some other type to a struct theorem

Fortunately, it is possible to statically analyze a C source file and confirm that it does none of these things, if such a level of assurance is desired.

TODO

Should really have even more demo non-proofs, to help ensure that it's not letting non-proofs pass themselves off as proofs somewhere.

History

1.1

• When creating a struct theorem or struct assumptions, the supplied struct formula * is now cloned (deep-copied) before being stored in the structure. This prevents the theorem from being manipulated after its creation by updating the contents of the struct formula * that was used to create it. Thanks to Proloy Mishra for pointing out this hole.
• More test cases, and a driver script (test.sh) to run the tests.
• Added this "History" section to the README.

1.0-2022.0905

• Renamed struct theorem to struct proof, and proof.{c,h} to theorem.{c,h} to better reflect what these objects are.

1.0

• Initial release of Philomath.